physics problems not solved yet

The 18 biggest unsolved mysteries in physics

Profound physics.

In 1900, the British physicist Lord Kelvin is said to have pronounced: "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement." Within three decades, quantum mechanics and Einstein's theory of relativity had revolutionized the field. Today, no physicist would dare assert that our physical knowledge of the universe is near completion. To the contrary, each new discovery seems to unlock a Pandora's box of even bigger, even deeper physics questions. These are our picks for the most profound open questions of all.

Related: Check out the 14 biggest historical mysteries that may never be solved.

Inside you’ll learn about parallel universes, why time seems to move in one direction only, and why we don’t understand chaos.

What is dark energy?

No matter how astrophysicists crunch the numbers, the universe simply doesn't add up. Even though gravity is pulling inward on space-time — the "fabric" of the cosmos — it keeps expanding outward faster and faster. To account for this, astrophysicists have proposed an invisible agent that counteracts gravity by pushing space-time apart. They call it dark energy . In the most widely accepted model of dark energy, it is a "cosmological constant": an inherent property of space itself, which has "negative pressure" driving space apart. As space expands, more space is created, and with it, more dark energy. Based on the observed rate of expansion, scientists know that the sum of all the dark energy must make up more than 70 percent of the total contents of the universe. But no one knows how to look for it. The best researchers have been able to do in recent years is narrow in a bit on where dark energy might be hiding, which was the topic of a study released in August 2015.

Next Up: Dark matter (scroll up to see the "Next" button)

What is dark matter?

Evidently, about 84 percent of the matter in the universe does not absorb or emit light. "Dark matter," as it is called, cannot be seen directly, and it hasn't yet been detected by indirect means, either. Instead, dark matter's existence and properties are inferred from its gravitational effects on visible matter, radiation and the structure of the universe. This shadowy substance is thought to pervade the outskirts of galaxies, and may be composed of "weakly interacting massive particles," or WIMPs. Worldwide, there are several detectors on the lookout for WIMPs, but so far, not one has been found. One recent study suggests dark mater might form long, fine-grained streams throughout the universe, and that such streams might radiate out from Earth like hairs. [Related: If Not Dark Matter, then What? ]

Next Up: Time's arrow

Why is there an arrow of time?

Time moves forward because a property of the universe called "entropy," roughly defined as the level of disorder, only increases, and so there is no way to reverse a rise in entropy after it has occurred. The fact that entropy increases is a matter of logic: There are more disordered arrangements of particles than there are ordered arrangements, and so as things change, they tend to fall into disarray. But the underlying question here is, why was entropy so low in the past? Put differently, why was the universe so ordered at its beginning, when a huge amount of energy was crammed together in a small amount of space? [ What's the Total Energy in the Universe? ]

Next Up: Parallel universes

Are there parallel universes?

Astrophysical data suggests space-time might be "flat," rather than curved, and thus that it goes on forever. If so, then the region we can see (which we think of as "the universe") is just one patch in an infinitely large "quilted multiverse." At the same time, the laws of quantum mechanics dictate that there are only a finite number of possible particle configurations within each cosmic patch (10^10^122 distinct possibilities). So, with an  infinite number of cosmic patches, the particle arrangements within them are forced to repeat — infinitely many times over.  This means there are infinitely many parallel universes: cosmic patches exactly the same as ours (containing someone exactly like you), as well as patches that differ by just one particle's position, patches that differ by two particles' positions, and so on down to patches that are totally different from ours.

Is there something wrong with that logic, or is its bizarre outcome true? And if it is true, how might we ever detect the presence of parallel universes? Check out this excellent perspective from 2015 that looks into what "infinite universes" would mean.

Next Up: Matter vs. Antimatter

Why is there more matter than antimatter?

The question of why there is so much more matter than its oppositely-charged and oppositely-spinning twin, antimatter, is actually a question of why anything exists at all. One assumes the universe would treat matter and antimatter symmetrically, and thus that, at the moment of the Big Bang, equal amounts of matter and antimatter should have been produced. But if that had happened, there would have been a total annihilation of both: Protons would have canceled with antiprotons, electrons with anti-electrons (positrons), neutrons with antineutrons, and so on, leaving behind a dull sea of photons in a matterless expanse. For some reason, there was excess matter that didn't get annihilated, and here we are. For this, there is no accepted explanation. The most detailed test to date of the differences between matter and antimatter, announced in August 2015, confirm they are mirror images of each other, providing exactly zero new paths toward understanding the mystery of why matter is far more common.

Next Up: Fate of the universe

What is the fate of the universe?

The fate of the universe strongly depends on a factor of unknown value: Ω, a measure of the density of matter and energy throughout the cosmos. If Ω is greater than 1, then space-time would be "closed" like the surface of an enormous sphere. If there is no dark energy, such a universe would eventually stop expanding and would instead start contracting, eventually collapsing in on itself in an event dubbed the "Big Crunch." If the universe is closed but there is dark energy, the spherical universe would expand forever.

Alternatively, if Ω is less than 1, then the geometry of space would be "open" like the surface of a saddle. In this case, its ultimate fate is the "Big Freeze" followed by the "Big Rip": first, the universe's outward acceleration would tear galaxies and stars apart, leaving all matter frigid and alone. Next, the acceleration would grow so strong that it would overwhelm the effects of the forces that hold atoms together, and everything would be wrenched apart.

If Ω = 1, the universe would be flat, extending like an infinite plane in all directions. If there is no dark energy, such a planar universe would expand forever but at a continually decelerating rate, approaching a standstill. If there is dark energy, the flat universe ultimately would experience runaway expansion leading to the Big Rip. Regardless how it plays out, the universe is dying, a fact discussed in detail by astrophysicist Paul Sutter in the essay from December, 2015.

Que sera, sera.

Next Up: An even stranger concept

How do measurements collapse quantum wavefunctions?

In the strange realm of electrons, photons and the other fundamental particles, quantum mechanics is law. Particles don't behave like tiny balls, but rather like waves that are spread over a large area. Each particle is described by a "wavefunction," or probability distribution, which tells what its location, velocity, and other properties are more likely to be, but not what those properties are. The particle actually has a range of values for all the properties, until you experimentally measure one of them — its location, for example — at which point the particle's wavefunction "collapses" and it adopts just one location. [ Newborn Babies Understand Quantum Mechanics ]

But how and why does measuring a particle make its wavefunction collapse, producing the concrete reality that we perceive to exist? The issue, known as the measurement problem, may seem esoteric, but our understanding of what reality is, or if it exists at all, hinges upon the answer. 

Next Up: String theory

Is string theory correct?

When physicists assume all the elementary particles are actually one-dimensional loops, or "strings," each of which vibrates at a different frequency, physics gets much easier. String theory allows physicists to reconcile the laws governing particles, called quantum mechanics, with the laws governing space-time, called general relativity, and to unify the four fundamental forces of nature into a single framework. But the problem is, string theory can only work in a universe with 10 or 11 dimensions: three large spatial ones, six or seven compacted spatial ones, and a time dimension. The compacted spatial dimensions — as well as the vibrating strings themselves — are about a billionth of a trillionth of the size of an atomic nucleus. There's no conceivable way to detect anything that small, and so there's no known way to experimentally validate or invalidate string theory.

Finally: We end with chaos . . .

Is there order in chaos?

Physicists can't exactly solve the set of equations that describes the behavior of fluids, from water to air to all other liquids and gases. In fact, it isn't known whether a general solution of the so-called Navier-Stokes equations even exists, or, if there is a solution, whether it describes fluids everywhere, or contains inherently unknowable points called singularities. As a consequence, the nature of chaos is not well understood. Physicists and mathematicians wonder, is the weather merely difficult to predict, or inherently unpredictable? Does turbulence transcend mathematical description, or does it all make sense when you tackle it with the right math ?

Congratulations on making it through this list of heavy topics. How about something lighter now? 25 Fun Facts in Science & History

Do the universe's forces merge into one?

The universe experiences four fundamental forces: electromagnetism, the strong nuclear force , the weak interaction (also known as the weak nuclear force ) and gravity . To date, physicists know that if you turn up the energy enough — for example, inside a particle accelerator — three of those forces "unify" and become a single force. Physicists have run particle accelerators and unified the electromagnetic force and weak interactions, and at higher energies, the same thing should happen with the strong nuclear force and, eventually, gravity.

But even though theories say that should happen, nature doesn't always oblige. So far, no particle accelerator has reached energies high enough to unify the strong force with electromagnetism and the weak interaction. Including gravity would mean yet more energy. It isn't clear whether scientists could even build one that powerful; the Large Hadron Collider (LHC), near Geneva, can send particles crashing into each other with energies in the trillions of electron volts (about 14 tera-electron volts, or TeV). To reach grand unification energies, particles would need at least a trillion times as much, so physicists are left to hunt for indirect evidence of such theories. 

Besides the issue of energies, Grand Unified Theories (GUTs) still have some problems because they predict other observations that so far haven't panned out. There are several GUTs that say protons, over immense spans of time (on the order of 10^36 years), should turn into other particles. This has never been observed, so either protons last much longer than anyone thought or they really are stable forever. Another prediction of some types of GUT is the existence of magnetic monopoles — isolated "north" and "south" poles of a magnet — and nobody has seen one of those, either. It's possible we just don't have a powerful enough particle accelerator. Or, physicists could be wrong about how the universe works. 

What happens inside a black hole?

What happens to an object's information if it gets sucked into a black hole ? According to the current theories, if you were to drop a cube of iron into a black hole, there would be no way to retrieve any of that information. That's because a black hole's gravity is so strong that its escape velocity is faster than light — and light is the fastest thing there is . However, a branch of science called quantum mechanics says that quantum information can't be destroyed. "If you annihilate this information somehow, something goes haywire," said Robert McNees, an associate professor of physics at Loyola University Chicago. [ How to Teleoport Info Out of a Black Hole ]

Quantum information is a bit different from the information we store as 1s and 0s on a computer, or the stuff in our brains. That's because quantum theories don't provide exact information about, for instance, where an object will be, like calculating the trajectory of a baseball in mechanics. Instead, such theories reveal the most likely location or the most likely result of some action. As a consequence, all of the probabilities of various events should add up to 1, or 100 percent. (For instance, when you roll a six-sided die, the chances of a given face coming up is one-sixth, so the probabilities of all the faces add up to 1, and you can't be more than 100 percent certain something will happen.) Quantum theory is, therefore, called unitary. If you know how a system ends, you can calculate how it began.

To describe a black hole, all you need is mass, angular momentum (if it's spinning) and charge. Nothing comes out of a black hole except a slow trickle of thermal radiation called Hawking radiation. As far as anyone knows, there's no way to do that reverse calculation to figure out what the black hole actually gobbled up. The information is destroyed. However, quantum theory says that information can't be completely out of reach. Therein lies the " information paradox ."

McNees said there has been a lot of work on the subject, notably by Stephen Hawking and Stephen Perry, who suggested in 2015 that, rather than being stored within the deep clutches of a black hole, the information remains on its boundary , called the event horizon. Many others have attempted to solve the paradox. Thus far, physicists can't agree on the explanation, and they're likely to disagree for some time. 

Do naked singularities exist?

A singularity occurs when some property of a "thing" is infinite, and so the laws of physics as we know them break down. At the center of black holes lies a point that is infinitely teensy and dense (packed with a finite amount of matter) — a point called a singularity. In mathematics , singularities come up all the time — dividing by zero is one instance, and a vertical line on a coordinate plane has an "infinite" slope. In fact, the slope of a vertical line is just undefined. But what would a singularity look like? And how would it interact with the rest of the universe? What does it mean to say that something has no real surface and is infinitely small?

A "naked" singularity is one that can interact with the rest of the universe. Black holes have event horizons — spherical regions from which nothing, not even light, can escape. At first glance, you might think the problem of naked singularities is partly solved for black holes at least, since nothing can get out of the event horizon and the singularity can't affect the rest of the universe. (It is "clothed," so to speak, while a naked singularity is a black hole without an event horizon.)

But whether singularities can form without an event horizon is still an open question. And if they can exist, then Albert Einstein's theory of general relativity will need a revision, because it breaks down when systems are too close to a singularity. Naked singularities might also function as wormholes , which would also be time machines — though there's no evidence for this in nature.

Violating charge-parity symmetry

If you swap a particle with its antimatter sibling, the laws of physics should remain the same. So, for example, the positively charged proton should look the same as a negatively charged antiproton. That's the principle of charge symmetry. If you swap left and right, again, the laws of physics should look the same. That's parity symmetry. Together, the two are called CP symmetry. Most of the time, this physics rule is not violated. However, certain exotic particles violate this symmetry. McNees said that's why it's strange. "There shouldn't be any violations of CP in quantum mechanics," he said. "We don't know why that is."

When sound waves make light

Though particle-physics questions account for many unsolved problems, some mysteries can be observed on a bench-top lab setup. Sonoluminescence is one of those. If you take some water and hit it with sound waves, bubbles will form. Those bubbles are low-pressure regions surrounded by high pressure; the outer pressure pushes in on the lower-pressure air, and the bubbles quickly collapse. When those bubbles collapse, they emit light, in flashes that last trillionths of a second.

The problem is, it's far from clear what the source of the light is. Theories range from tiny nuclear fusion reactions to some type of electrical discharge, or even compression heating of the gases inside the bubbles. Physicists have measured high temperatures inside these bubbles, on the order of tens of thousands of degrees Fahrenheit, and taken numerous pictures of the light they produce. But there's no good explanation of how sound waves create these lights in a bubble.

What lies beyond the Standard Model?

The Standard Model is one of the most successful physical theories ever devised. It's been standing up to experiments to test it for four decades, and new experiments keep showing that it is correct. The Standard Model describes the behavior of the particles that make up everything around us, as well as explaining why, for example, particles have mass. In fact, the discovery of the Higgs boson — a particle that gives matter its mass — in 2012 was a historic milestone because it confirmed the long-standing prediction of its existence. 

But the Standard Model doesn't explain everything. The Standard Model has made many successful predictions — for example, the Higgs boson, the W and Z boson (which mediate the weak interactions that govern radioactivity), and quarks among them — so it is difficult to see where physics might go beyond it. That said, most physicists agree that the Standard Model is not complete. There are several contenders for new, more complete models — string theory is one such model — but so far, none of these have been conclusively verified by experiments. 

Fundamental constants

Dimensionless constants are numbers that don't have units attached to them. The speed of light, for example, is a fundamental constant measured in units of meters per second (or 186,282 miles per second). Unlike the speed of light, dimensionless constants have no units and they can be measured, but they can't be derived from theories, whereas constants like the speed of light can be.

In his book "Just Six Numbers: The Deep Forces That Shape the Universe" (Basic Books, 2001), astronomer Martin Rees focuses on certain "dimensionless constants" he considers fundamental to physics. In fact, there are many more than six; about 25 exist in the Standard Model. [ The 9 Most Massive Numbers in Existence ]

For example, the fine structure constant, usually written as alpha, governs the strength of magnetic interactions. It is about 0.007297. What makes this number odd is that if it were any different, stable matter wouldn't exist. Another is the ratio of the masses of many fundamental particles, such as electrons and quarks, to the Planck mass (which is 1.22 ´10 19 GeV/c 2 ). Physicists would love to figure out why those particular numbers have the values they do, because if they were very different, the universe's physical laws wouldn't allow for humans to be here. And yet there's still no compelling theoretical explanation for why they have those values. 

What the heck is gravity, anyway?

What is gravity , anyway? Other forces are mediated by particles. Electromagnetism, for example, is the exchange of photons. The weak nuclear force is carried by W and Z bosons, and gluons carry the strong nuclear force that holds atomic nuclei together. McNees said all of the other forces can be quantized, meaning they could be expressed as individual particles and have noncontinuous values.

Gravity doesn't seem to be like that. Most physical theories say it should be carried by a hypothetical massless particle called a graviton. The problem is, nobody has found gravitons yet, and it's not clear that any particle detector that could be built could see them, because if gravitons interact with matter, they do it very, very rarely — so seldom that they'd be invisible against the background noise. It isn't even clear that gravitons are massless, though if they have a mass at all, it's very, very small — smaller than that of neutrinos, which are among the lightest particles known. String theory posits that gravitons (and other particles) are closed loops of energy, but the mathematical work hasn't yielded much insight so far.

Because gravitons haven't been observed yet, gravity has resisted attempts to understand it in the way we understand other forces – as an exchange of particles. Some physicists, notably Theodor Kaluza and Oskar Klein, posited that gravity may be operating as a particle in extra dimensions beyond the three of space (length, width, and height) and one of time (duration)we are familiar with, but whether that is true is still unknown. 

Do we live in a false vacuum?

The universe seems relatively stable. After all, it's been around for about 13.8 billion years . But what if the whole thing were a massive accident?

It all starts with the Higgs and the universe's vacuum. Vacuum, or empty space, should be the lowest possible energy state, because there's nothing in it. Meanwhile, the Higgs boson — via the so-called Higgs field — gives everything its mass. Writing in the journal Physics, Alexander Kusenko, a professor of physics and astronomy at the University of California, Los Angeles, said the energy state of the vacuum can be calculated from the potential energy of the Higgs field and the masses of the Higgs and top quark (a fundamental particle).

So far, those calculations appear to show that the universe's vacuum might not be in the lowest possible energy state. That would mean it's a false vacuum. If that's true, our universe might not be stable, because a false vacuum can be knocked into a lower energy state by a sufficiently violent and high-energy event. If that were to happen, there would be a phenomenon called bubble nucleation. A sphere of lower-energy vacuum would start growing at the speed of light. Nothing, not even matter itself, would survive. Effectively, we'd be replacing the universe with another one, which might have very different physical laws. [ 5 Reasons We May Live in a Multiverse ]

That sounds scary, but given that the universe is still here, clearly there hasn't been such an event yet, and astronomers have seen gamma-ray bursts , supernovas, and quasars, all of which are pretty energetic. So it's probably unlikely enough that we wouldn't need to worry. That said, the idea of a false vacuum means that our universe might have popped into existence in just that way, when a previous universe's false vacuum was knocked into a lower energy state. Perhaps we were the result of an accident with a particle accelerator. 

Editor's note: This list was originally published in 2012. It was updated on Feb. 27, 2017, to include newer information and recent studies.

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10 mysteries that physics can't answer... yet

From why we travel forwards in time to how bicycles travel forwards at all, we present the questions great and small that our finest minds can't explain

What came before the big bang?

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Is the universe infinite or just very big, how long does a proton live, why is ice slippery, what is glass, can we get energy from nothing, why does space have three dimensions, why do we move forwards in time, where does quantum weirdness end, how creative computers will dream up things we'd never imagine.

The 7 Biggest Unanswered Questions in Physics

If Isaac Newton suddenly popped out of a time machine, he’d be delighted to see how far physics had come. Things that were deeply mysterious a few centuries ago are now taught in freshman physics classes (the composition of stars is one good example).

Newton would be stunned to see enormous experiments like the Large Hadron Collider (LHC) in Switzerland — and possibly perturbed to learn that his theory of gravity had been superseded by one dreamed up by some fellow named Einstein. Quantum mechanics would probably strike him as bizarre, though today’s scientists feel the same way.

But once he was up to speed, Newton would no doubt applaud what modern physics has achieved — from the discovery of the nature of light in the 19th Century to determining the structure of the atom in the 20th Century to last year’s discovery of gravitational waves . And yet physicists today are the first to admit they don’t have all the answers. “There are basic facts about the universe that we’re ignorant of,” says Dr. Daniel Whiteson, a University of California physicist and the co-author of the new book "We Have No Idea: A Guide to the Unknown Universe."

What follows is a brief tour through seven of the biggest unsolved problems in physics. (If you’re wondering why head-scratchers like dark matter and dark energy aren’t on the list, it’s because they were in our earlier story on the Five Biggest Questions about the Universe .)

1. What is matter made of?

We know matter is made up atoms, and atoms are made up of protons, neutrons, and electrons. And we know that protons and neutrons are made up of smaller particles known as quarks. Would probing deeper uncover particles even more fundamental? We don't know for sure.

We do have something called the Standard Model of particle physics, which is very good at explaining the interactions between subatomic particles. The Standard Model has also been used to predict the existence of previously unknown particles. The last particle to be found this way was the Higgs boson , which LHC researchers discovered in 2012.

But there’s a hitch.

“The Standard Model doesn’t explain everything,” says Dr. Don Lincoln, a particle physicist at Fermi National Accelerator Laboratory (Fermilab) near Chicago. “It doesn’t explain why the Higgs boson exists. It doesn’t explain in detail why the Higgs boson has the mass that it does.” In fact, the Higgs turned out to be a heck of a lot less massive than predicted — theory had held that it would be about “a quadrillion times heavier than it is,” says Lincoln.

The mysteries don’t end there. Atoms are known to be electrically neutral — the positive charge of the protons is cancelled out by the negative charge of the electrons — but as to why this is so, Lincoln says, “Nobody knows.”

2. Why is gravity so weird?

No force is more familiar than gravity — it’s what keeps our feet on the ground, after all. And Einstein’s theory of general relativity gives a mathematical formulation for gravity, describing it as a “warping” of space. But gravity is a trillion trillion trillion times weaker than the other three known forces (electromagnetism and the two kinds of nuclear forces that operate over tiny distances).

One possibility — speculative at this point — is that in addition to the three dimensions of space that we notice every day, there are hidden extra dimensions , perhaps “curled up” in a way that makes them impossible to detect. If these extra dimensions exist — and if gravity is able to “leak” into them — it could explain why gravity seems so weak to us.

“It could be that gravity is as strong as these other forces but that it gets rapidly diluted by spilling out into these other invisible dimensions,” says Whiteson. Some physicists hoped that experiments at the LHC would give a hint of these extra dimensions — but so far, no luck.

3. Why does time seem to flow only in one direction?

Since Einstein, physicists have thought of space and time as forming a four-dimensional structure known as “spacetime.” But space differs from time in some very fundamental ways. In space, we’re free to move about as we wish. When it comes to time, we’re stuck. We grow older, not younger. And we remember the past, but not the future. Time, unlike space, seems to have a preferred direction — physicists call it the “arrow of time.”

Some physicists suspect that the second law of thermodynamics provides a clue. It states that the entropy of a physical system (roughly, the amount of disorder) rises over time, and physicists think this increase is what gives time its direction. (For example, a broken teacup has more entropy than an intact one — and, sure enough, smashed teacups always seem to arise after intact ones, not before.)

Entropy may be rising now because it was lower earlier, but why was it low to begin with? Was the entropy of the universe unusually low 14 billion years ago, when the Big Bang brought it into existence?

For some physicists, including Caltech’s Sean Carroll, that’s the missing piece of the puzzle. “If you can tell me why the early universe had a low entropy, then I can explain the rest of it,” he says. In Whiteson’s view, entropy isn't the whole story. “To me,” he says, “the deepest part of the question is, why is time so different from space?” (Recent computer simulations seem to show how the asymmetry of time might arise from the fundamental laws of physics, but the work is controversial, and the ultimate nature of time continues to stir passionate debate .)

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4. where did all the antimatter go.

Antimatter may be more famous in fiction than in real life. On the original Star Trek, antimatter reacts with ordinary matter to power the warp drive that propels the U.S.S. Enterprise at faster-than-light velocities. While warp drive is pure fiction, antimatter is very real. We know that for each particle of ordinary matter, it's possible to have an identical particle with the opposite electrical charge. An antiproton is just like a proton, for example, but with a negative charge. The antiparticle corresponding to the negatively charged electron, meanwhile, is the positively charged positron.

Physicists have created antimatter in the laboratory. But when they do, they create an equal amount of matter. That suggests that the Big Bang must have created matter and antimatter in equal quantities. Yet almost everything we see around us, from the ground beneath our feet to the most remote galaxies, is made of ordinary matter.

What’s going on? Why is there more matter than antimatter? Our best guess is that the Big Bang somehow produced a tiny bit more matter than antimatter. “What had to have happened early in the history of the universe — in the very moments after the Big Bang — is that for every 10 billion antimatter particles there were 10 billion and one matter particles,” says Lincoln. “And the matter and the antimatter annihilated the 10 billion, leaving the one. And that little ‘one’ is the mass that makes up us.”

But why the slight excess of matter over antimatter in the first place? “We really don’t understand that,” Lincoln says. “It’s bizarre.” Had the initial amounts of matter and antimatter been equal, they’d have annihilated each other completely in a burst of energy. In which case, says Lincoln, “we wouldn’t exist.”

Some answers may come when the Deep Underground Neutrino Experiment (DUNE) starts collecting data in 2026. DUNE will analyze a beam of neutrinos — tiny, chargeless and nearly massless particles — fired from Fermilab to the Sanford Underground Research Facility in South Dakota, some 800 miles away. The beam will include neutrinos and antineutrinos, with the aim of seeing if they behave in the same manner — thus potentially providing a clue to nature’s matter-antimatter asymmetry.

5. What happens in the gray zone between solid and liquid?

Solids and liquids are well understood. But some materials act like both a liquid and a solid, making their behavior hard to predict. Sand is one example. A grain of sand is as solid as a rock, but a million grains can flow through a funnel almost like water. And highway traffic can behave in a similar way, flowing freely until it becomes blocked at some bottleneck.

So a better understanding of this “gray zone” might have important practical applications.

“People have been asking, under what conditions does the entire system jam up or clog?” says Dr. Kerstin Nordstrom, a physicist at Mount Holyoke College. “What are the crucial parameters to avoid clogging?” Weirdly, an obstruction in the flow of traffic can, under certain conditions, actually reduce traffic jams. “It’s very counterintuitive,” she says.

6. Can we find a unified theory of physics?

We now have two overarching theories to explain just about every physical phenomenon: Einstein’s theory of gravity (general relativity) and quantum mechanics. The former is good at explaining the motion of everything from golf balls to galaxies. Quantum mechanics is equally impressive in its own domain — the realm of atoms and subatomic particles.

Trouble is, the two theories describe our world in very different terms. In quantum mechanics, events unfold against a fixed backdrop of spacetime — while in general relativity, spacetime itself is flexible. What would a quantum theory of curved space-time look like? We don’t know, says Carroll. “We don’t even know what it is we’re trying to quantize.”

That hasn’t stopped people from trying. For decades now, string theory — which pictures matter as made up of tiny vibrating strings or loops of energy — has been touted as the best bet for producing a unified theory of physics. But some physicists prefer loop quantum gravity , in which space itself is imagined to be made of tiny loops.

Each approach has enjoyed some success — techniques developed by string theorists, in particular, are proving useful for tackling certain difficult physics problems. But neither string theory or loop quantum gravity has been tested experimentally. For now, the long-sought “theory of everything” continues to elude us.

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7. how did life evolve from nonliving matter.

For its first half-billion years, Earth was lifeless. Then life took hold, and it has thrived ever since. But how did life arise? Before biological evolution began, scientists believe there was chemical evolution , with simple inorganic molecules reacting to form complex organic molecules, most likely in the oceans. But what kick-started this process in the first place?

MIT physicist Dr. Jeremy England recently put forward a theory that attempts to explain the origin of life in terms of fundamental principles of physics. In this view, life is the inevitable result of rising entropy. If the theory is correct, the arrival of life “should be as unsurprising as rocks rolling downhill,” England told Quanta magazine in 2014.

The idea is highly speculative. Recent computer simulations , however, may be lending support to it. The simulations show that ordinary chemical reactions (of the sort that would have been common on the newly formed Earth) can lead to the creation of highly structured compounds — seemingly a crucial stepping-stone on the path to living organisms.

What makes life so hard for physicists to study? Anything that’s alive is “far from equilibrium,” as a physicist would put it. In a system in equilibrium, one component is pretty much like every other, with no flow of energy in or out. (A rock would be an example; a box full of gas is another.) Life is just the opposite. A plant, for example, absorbs sunlight and uses its energy to make complex sugar molecules while radiating heat back into the environment.

Understanding these complex systems “is the great unsolved problem in physics,” says Stephen Morris, a University of Toronto physicist. “How do we deal with these far-from-equilibrium systems which self-organize into amazing, complex things — like life?”

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Why students still can't solve physics problems after solving over 2000 problems

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Taejin Byun , Gyoungho Lee; Why students still can't solve physics problems after solving over 2000 problems. American Journal of Physics 1 September 2014; 82 (9): 906–913. https://doi.org/10.1119/1.4881606

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This study investigates the belief that solving a large number of physics problems helps students better learn physics. We investigated the number of problems solved, student confidence in solving these problems, academic achievement, and the level of conceptual understanding of 49 science high school students enrolled in upper-level physics classes from Spring 2010 to Summer 2011. The participants solved an average of 2200 physics problems before entering high school. Despite having solved so many problems, no statistically significant correlation was found between the number of problems solved and academic achievement on either a mid-term or physics competition examination. In addition, no significant correlation was found between the number of physics problems solved and performance on the Force Concept Inventory (FCI). Lastly, four students were selected from the 49 participants with varying levels of experience and FCI scores for a case study. We determined that their problem solving and learning strategies was more influential in their success than the number of problems they had solved.

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10 Unsolved Problems In Astrophysics That Are Way Too Interesting.

Admin and Founder of ‘The Secrets Of The Universe’ and former intern at Indian Institute of Astrophysics, Bangalore, I am a science student pursuing a Master’s in Physics from India . I love to study and write about Stellar Astrophysics, Relativity & Quantum Mechanics.

Though it has been more than 200 years since astrophysics formally began, what we know is just a drop. The vast ocean of knowledge lays undiscovered before us. I thought of ending this series by writing about some of the major unsolved problems in astrophysics.

Exactly two months ago, we started our journey of exploring the cosmos. We started with the most basic question:  What is Astrophysics? We learned some of the basic tools used in this subject: the EM Spectrum ,  distances in astronomy , the concept of magnitude , the classification of stars , types of redshifts , basics of telescopes , the Hertzsprung Russell Diagram, and so on. Then we studied the structure of Sun ,  nuclear reactions in stars ,  stellar evolution , the formation of neutron stars and black holes,   types of black holes ,  quasars ,  galaxies ,  nebulae , dark matter , gravitational waves , radio astronomy , and the CMB radiation. We shall now conclude this series by discussing the top 10 unsolved problems in Astrophysics.

Read all the articles of Basics of Astrophysics series here

Unsolved Problems In Astrophysics

10. tabby’s star.

Tabby’ star, formally known as KIC 8462852, is an F type main-sequence star in the constellation of Cygnus. What brought Tabby’s star in limelight is its anomalous dimming. One of the graphs that shows the maximum dimming of around 22% is given below.

Several hypotheses have been put forward to explain these fluctuations. The first one says that the star is surrounded by an uneven ring of dust. According to NASA, researchers found less dimming in the infrared light from the star than in its ultraviolet light. Any object larger than dust particles would dim all wavelengths of light equally when passing in front of Tabby’s Star. The second hypothesis says that Tabby’s star is surrounded by a cloud of disintegrating comets orbiting the star elliptically. But, the fact that a cloud of a comet can cause dimming up to 22% has cast doubt on this theory.

Another interesting hypothesis says that the dimming might be a result of an alien megastructure. That alien megastructure could be the Dyson swarm, built by civilizations to intercept their star’s light for their energy needs. However, the likelihood of extraterrestrial intelligence being the cause of the dimming is very low

9. Origin of Magnetar Magnetic Field

Magnetars are neutron stars with a tremendous magnetic field. The magnetic field of a typic magnetar lies between 1-100 billion Tesla. To bring it to perspective, the maximum magnetic field that can be generated under special laboratory conditions is just a few hundred Tesla. Just like the neutron stars, they are about 20 Km wide and have a mass of 2-3 times that of Sun. This implies they are quite dense. A tablespoon of its material will weigh 100 million tonnes.

The origin of such a strong magnetic field is hypothesized to be a magnetohydrodynamic process in the turbulent, extremely dense conducting fluid that exists before the neutron star settles into its equilibrium position. The magnetic field of a magnetar would be lethal even at a distance of 1000 km due to the strong magnetic field distorting the electron clouds of the subject’s constituent atoms, rendering the chemistry of life impossible. At a distance of halfway from Earth to the moon, a magnetar could strip information from the magnetic stripes of all credit cards on Earth. As of 2010, they are the most powerful magnetic objects detected throughout the universe.

Popular in series:

• How do neutron stars and black holes form?
• The most powerful object in the universe
• Gravitational waves and their importance

8. Fast Radio Bursts

Fast Radio Bursts (FRBs) are unresolved (point source-like), broadband (spanning a large range of radio frequencies), millisecond flashes found in parts of the sky. The physical phenomenon that causes these bursts is still a mystery. Possible sources of FRBs are neutron stars, black holes, or extraterrestrial intelligence.

Although the exact origin and cause is uncertain, most are believed to be extragalactic. The first Milky Way FRB was detected in April 2020. The origin of FRBs is still one of the most intriguing unsolved problems in astrophysics.

7. Ultra-High Energy Cosmic Rays

The Ultra-High Energy Cosmic Rays (UHECR) are the cosmic rays with unimaginably high energy: greater than exa electron volt (10^18 eV). The Oh My God particle by the University of Utah’s Fly’s Eye experiment on the evening of 15 October 1991 over Dugway Proving Ground, Utah was a shock to astrophysicists.

They estimated its energy to be approximately 3.2×10^20 eV (50 J) —in other words, an atomic nucleus with kinetic energy equal to that of a baseball (5 ounces or 142 grams) traveling at about 100 kilometers per hour (60 mph). The origin of such particles is still a hypothesis. It is one of the major unsolved problems in astrophysics.

Recommended Book To Start Astrophysics

6. the solar cycle.

The solar cycle is a repeated cycle of solar fluctuations that has a period of 11 years. Every 11 years, the magnetic field of the Sun flips completely. This means that the Sun’s north and south poles switch places. The solar activity is also influenced by this cycle. The beginning of the solar cycle is the solar minimum, with minimum sunspots. Then, in the middle of the cycle, the solar activity reaches its maximum and the number of sunspots also increases.

Giant eruptions on the Sun, such as solar flares and coronal mass ejections, also increase during the solar cycle. These eruptions send powerful bursts of energy and material into space. Understanding the solar cycle is still a great mystery for scientists.

5. The Corona Mystery

Another unsolved problem in solar physics is the corona mystery. The corona is the outermost part of the solar atmosphere. It can be seen during a total solar eclipse. The problem is that the corona has a temperature of about a million K while the surface of the Sun, the photosphere, is at about 5,900 K. How can the heat flow from cold to the hot body? How can the most basic law of thermodynamics break down? Is there any other mechanism that is taking place in the corona? If yes, then what is it? Is it the Alfven Waves?

4. The Lithium Problem

The lithium problem is an astrophysical problem on the abundance of Li-7 isotope. Minutes after the big bang, the first elements formed. These included hydrogen, helium, lithium, and trace amounts of other elements. The observed composition of the universe is consistent with the big bang model for hydrogen and helium. However, there is a discrepancy when it comes to lithium. The most widely accepted models of the Big Bang suggest that three times as much primordial lithium should exist.

Several solutions have been hypothesized to solve the missing lithium problem. There might be a need for more accurate determination of the abundance of lithium in the universe. The astrophysical solution says that there is an error in calculation. Another possible way to find the remaining lithium is to make corrections in nuclear physics. Incorrect or missing reactions could give rise to the lithium problem.

3. Hawking Radiation

The Hawking radiation is electromagnetic radiation that is predicted to be released by black holes due to quantum events near the event horizon.

It is important to recall that for all discovered and localized black holes, this effect is too small to be measured under experimentally achievable conditions. Because of that, Hawking radiation has not been detected yet. In the meantime, physicists are building and analyzing analogous systems.

In September 2010, it was thought that a laboratory-created body, simulating a white hole’s event horizon radiated an optical analog to Hawking radiation, although to this date no official confirmation of the accuracy of this experiment exists. One of the latest predictions also links sonic black holes (for which sound perturbations are analogous to light in a gravitational black hole) to a form of perfect fluid flow that could simulate Hawking radiation. Read about Hawking radiation in our detailed article here.

You may also like:

• How to become an astrophysicist?
• Top 10 astrophysicists in history
• Best books on astrophysics that will help you decode the universe

While it is not yet known how and when will Hawking radiation be detected, scientists are diligently working towards this purpose. What happens after such radiation is discovered is also unknown, but one can only hope that it will open a new era when it comes to understanding black holes. 

2. Galaxy Rotation Curve And Dark Matter

As we studied in detail in the article on the dark universe , A galactic rotation curve is the plot of the orbital rotational velocity of stars versus their distance from the center. Consider the solar system. Mercury orbits the Sun in 88 days while for Neptune, it takes about 165 years. Also, the orbital velocity of planets decreases as we go from Mercury to Neptune. However, this is not true for disk galaxies. Stars revolve around their galaxy’s center at equal or increasing speed over a large range of distances.

This discrepancy has two implications. Either Newton’s laws of classical mechanics aren’t universal or there is an additional matter in the galaxy that isn’t visible to us. This invisible matter is known as dark matter. The discrepancy in the galaxy rotation curves is one of the first evidence of dark matter.

How was the series?

1. black holes.

One of the most important unsolved problems in Astrophysics is black holes. Black holes first appeared in the solutions of the equations of general relativity. Though a lot of research is taking place in this field, the question is: Do the mathematical black holes predicted by General Relativity really exist, or are they the eternally collapsing objects?

Also watch: What is string theory?

The scientific community is divided into two sects: one which says that the black holes observed are the ones that are predicted by GTR, having a singularity while others say that they aren’t the mathematical black holes but are eternally collapsing objects. There are many observed facts that the former cannot explain. One of them is the strength of the magnetic field. How can black holes produce such a strong magnetic field when the only source is the particles of the accretion disk. You can find peer-reviewed research papers on the Eternally Collapsing Objects here .

The model of eternally collapsing objects instead of black holes accounts for many observed facts and is thought to be a better model. But still far away from being accepted worldwide.

These were some of the major unsolved problems in astrophysics.

Author’s Message

This concludes the Basics of Astrophysics series. These 30 articles were one of the most challenging projects taken up by us. It took us a lot of time to design, write, and share this series with you. Squeezing such a vast topic into just 30 easy to understand articles was difficult. I am glad that the series received a great response worldwide. One of the basic purpose to bring such a series was to tell the importance of physics in this field. I also wanted to shed light on the vastness of the subject. People think that Astrophysics is all about the glamorous topics, the ones popular in science fiction. But no, there is much more and these 30 articles are proof.

Though the series is ending, the journey will continue. Our team will be back with another educational series like this one. Keep visiting us.

Hello Rishab, this article was very interesting and informative. And I also came to know about many windows that are open in the research field. I really enjoyed reading your article on basics of astrophysics. thank you .

Dear Rishab, I have thoroughly enjoyed reading your articles. A big compliment for the effort you have put into it, especially the Q&A videos provided a lot of extra information. I am a complete amateur to this field of science (also 63 years old, studied shipbuilding when I was younger and I am not going to take up physics/mathematics again to become an astrophysicist) but it was a nice journey to freshen up or learn about new topics you and your co-writers presented in these articles. All I have to do now is to dive deeper into the theory by exploring all the pingbacks that you linked to in the comment sections. A lot more interesting reading material lies ahead of me, thanks again!

[…] 10 unsolved problems in Astrophysics […]

[…] Read Now […]

[…] 10 unsolved problems in astrophysics […]

I am a curious student right now studying in 12th science. I am very interested in Astronomy (i.e. everything related to space for me?) from always. I am reading all your articles and always excited to know more. Thank you for all this.?

This is really a good article. I’m a student from Kolkata, India pursuing my B.Sc. physics from Calcutta University and dream of doing cosmology in the future I like your blog very much. And I’m also trying have a blog of my own here https://naturescience234.blogspot.com

i read this and i am shocked how they dont even introduce us to basic astrophysics in high school

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How to Solve Any Physics Problem

Last Updated: December 20, 2022

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 23 people, some anonymous, worked to edit and improve it over time. This article has been viewed 304,259 times. Learn more...

Baffled as to where to begin with a physics problem? There is a very simply and logical flow process to solving any physics problem.

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• Many people report that if they leave a problem for a while and come back to it later, they find they have a new perspective on it and can sometimes see an easy way to the answer that they did not notice before. ⧼thumbs_response⧽ Helpful 247 Not Helpful 46
• Try to understand the problem first. ⧼thumbs_response⧽ Helpful 185 Not Helpful 49
• Remember, the physics part of the problem is figuring out what you are solving for, drawing the diagram, and remembering the formulae. The rest is just use of algebra, trigonometry, and/or calculus, depending on the difficulty of your course. ⧼thumbs_response⧽ Helpful 113 Not Helpful 34

• Physics is not easy to grasp for many people, so do not get bent out of shape over a problem. ⧼thumbs_response⧽ Helpful 98 Not Helpful 24
• If an instructor tells you to draw a free body diagram, be sure that that is exactly what you draw. ⧼thumbs_response⧽ Helpful 87 Not Helpful 24

Things You'll Need

• A Writing Utensil (preferably a pencil or erasable pen of sorts)
• Calculator with all the functions you need for your exam
• An understanding of the equations needed to solve the problems. Or a list of them will suffice if you are just trying to get through the course alive.

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• ↑ https://iopscience.iop.org/article/10.1088/1361-6404/aa9038
• ↑ https://physics.wvu.edu/files/d/ce78505d-1426-4d68-8bb2-128d8aac6b1b/expertapproachtosolvingphysicsproblems.pdf
• ↑ https://www2.southeastern.edu/Academics/Faculty/rallain/scaleup/problem_solving.html
• ↑ https://www.brighthubeducation.com/science-homework-help/42596-tips-to-choosing-the-correct-physics-formula/

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10 Hard Math Problems That Continue to Stump Even the Brightest Minds

Maybe you’ll have better luck.

For now, you can take a crack at the hardest math problems known to man, woman, and machine. ✅ More from Popular Mechanics :

• Euler’s Number Is Seriously Everywhere. Here’s What Makes It So Special
• Fourier Transforms: The Math That Made Color TV Possible
• The Game of Trees is a Mad Math Theory That Is Impossible to Prove

The Collatz Conjecture

In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is promising, the problem isn’t fully solved yet.

A refresher on the Collatz Conjecture : It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity).

✅ Down the Rabbit Hole: The Math That Helps the James Webb Space Telescope Sit Steady in Space

Tao’s recent work is a near-solution to the Collatz Conjecture in some subtle ways. But he most likely can’t adapt his methods to yield a complete solution to the problem, as Tao subsequently explained. So, we might be working on it for decades longer.

The Conjecture lives in the math discipline known as Dynamical Systems , or the study of situations that change over time in semi-predictable ways. It looks like a simple, innocuous question, but that’s what makes it special. Why is such a basic question so hard to answer? It serves as a benchmark for our understanding; once we solve it, then we can proceed onto much more complicated matters.

The study of dynamical systems could become more robust than anyone today could imagine. But we’ll need to solve the Collatz Conjecture for the subject to flourish.

Goldbach’s Conjecture

One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude. But we need proof for all natural numbers.

Goldbach’s Conjecture precipitated from letters in 1742 between German mathematician Christian Goldbach and legendary Swiss mathematician Leonhard Euler , considered one of the greatest in math history. As Euler put it, “I regard [it] as a completely certain theorem, although I cannot prove it.”

✅ Dive In: The Math Behind Our Current Theory of Human Color Perception Is Wrong

Euler may have sensed what makes this problem counterintuitively hard to solve. When you look at larger numbers, they have more ways of being written as sums of primes, not less. Like how 3+5 is the only way to break 8 into two primes, but 42 can broken into 5+37, 11+31, 13+29, and 19+23. So it feels like Goldbach’s Conjecture is an understatement for very large numbers.

Still, a proof of the conjecture for all numbers eludes mathematicians to this day. It stands as one of the oldest open questions in all of math.

The Twin Prime Conjecture

Together with Goldbach’s, the Twin Prime Conjecture is the most famous in Number Theory—or the study of natural numbers and their properties, frequently involving prime numbers. Since you've known these numbers since grade school, stating the conjectures is easy.

When two primes have a difference of 2, they’re called twin primes. So 11 and 13 are twin primes, as are 599 and 601. Now, it's a Day 1 Number Theory fact that there are infinitely many prime numbers. So, are there infinitely many twin primes? The Twin Prime Conjecture says yes.

Let’s go a bit deeper. The first in a pair of twin primes is, with one exception, always 1 less than a multiple of 6. And so the second twin prime is always 1 more than a multiple of 6. You can understand why, if you’re ready to follow a bit of heady Number Theory.

✅ Keep Learning: If We Draw Graphs Like This, We Can Change Computers Forever

All primes after 2 are odd. Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue. If a number is 3 more than a multiple of 6, then it has a factor of 3. Having a factor of 3 means a number isn’t prime (with the sole exception of 3 itself). And that's why every third odd number can't be prime.

How’s your head after that paragraph? Now imagine the headaches of everyone who has tried to solve this problem in the last 170 years.

The good news is that we’ve made some promising progress in the last decade. Mathematicians have managed to tackle closer and closer versions of the Twin Prime Conjecture. This was their idea: Trouble proving there are infinitely many primes with a difference of 2? How about proving there are infinitely many primes with a difference of 70,000,000? That was cleverly proven in 2013 by Yitang Zhang at the University of New Hampshire.

For the last six years, mathematicians have been improving that number in Zhang’s proof, from millions down to hundreds. Taking it down all the way to 2 will be the solution to the Twin Prime Conjecture. The closest we’ve come —given some subtle technical assumptions—is 6. Time will tell if the last step from 6 to 2 is right around the corner, or if that last part will challenge mathematicians for decades longer.

The Riemann Hypothesis

Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems , with $1 million reward for its solution. It has implications deep into various branches of math, but it’s also simple enough that we can explain the basic idea right here.

There is a function, called the Riemann zeta function, written in the image above.

For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using the imaginary number 𝑖—finding 𝜁(s) gets tricky.

So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers, this means the function has a certain behavior along a special vertical line. The hypothesis is that the behavior continues along that line infinitely.

✅ Stay Curious: How to Paint a Room Using Math

The Hypothesis and the zeta function come from German mathematician Bernhard Riemann, who described them in 1859. Riemann developed them while studying prime numbers and their distribution. Our understanding of prime numbers has flourished in the 160 years since, and Riemann would never have imagined the power of supercomputers. But lacking a solution to the Riemann Hypothesis is a major setback.

If the Riemann Hypothesis were solved tomorrow, it would unlock an avalanche of further progress. It would be huge news throughout the subjects of Number Theory and Analysis. Until then, the Riemann Hypothesis remains one of the largest dams to the river of math research.

The Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is another of the six unsolved Millennium Prize Problems, and it’s the only other one we can remotely describe in plain English. This Conjecture involves the math topic known as Elliptic Curves.

When we recently wrote about the toughest math problems that have been solved , we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir Andrew Wiles solved it using Elliptic Curves. So, you could call this a very powerful new branch of math.

✅ The Latest: Mathematicians Discovered Something Mind-Blowing About the Number 15

In a nutshell, an elliptic curve is a special kind of function. They take the unthreatening-looking form y²=x³+ax+b. It turns out functions like this have certain properties that cast insight into math topics like Algebra and Number Theory.

British mathematicians Bryan Birch and Peter Swinnerton-Dyer developed their conjecture in the 1960s. Its exact statement is very technical, and has evolved over the years. One of the main stewards of this evolution has been none other than Wiles. To see its current status and complexity, check out this famous update by Wells in 2006.

The Kissing Number Problem

A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to the idea of stacking many spheres in a given space, like fruit at the grocery store. Some questions in this study have full solutions, while some simple ones leave us stumped, like the Kissing Number Problem.

When a bunch of spheres are packed in some region, each sphere has a Kissing Number, which is the number of other spheres it’s touching; if you’re touching 6 neighboring spheres, then your kissing number is 6. Nothing tricky. A packed bunch of spheres will have an average kissing number, which helps mathematically describe the situation. But a basic question about the kissing number stands unanswered.

✅ Miracles Happen: Mathematicians Finally Make a Breakthrough on the Ramsey Number

First, a note on dimensions. Dimensions have a specific meaning in math: they’re independent coordinate axes. The x-axis and y-axis show the two dimensions of a coordinate plane. When a character in a sci-fi show says they’re going to a different dimension, that doesn’t make mathematical sense. You can’t go to the x-axis.

A 1-dimensional thing is a line, and 2-dimensional thing is a plane. For these low numbers, mathematicians have proven the maximum possible kissing number for spheres of that many dimensions. It’s 2 when you’re on a 1-D line—one sphere to your left and the other to your right. There’s proof of an exact number for 3 dimensions, although that took until the 1950s.

Beyond 3 dimensions, the Kissing Problem is mostly unsolved. Mathematicians have slowly whittled the possibilities to fairly narrow ranges for up to 24 dimensions, with a few exactly known, as you can see on this chart . For larger numbers, or a general form, the problem is wide open. There are several hurdles to a full solution, including computational limitations. So expect incremental progress on this problem for years to come.

The Unknotting Problem

The simplest version of the Unknotting Problem has been solved, so there’s already some success with this story. Solving the full version of the problem will be an even bigger triumph.

You probably haven’t heard of the math subject Knot Theory . It ’s taught in virtually no high schools, and few colleges. The idea is to try and apply formal math ideas, like proofs, to knots, like … well, what you tie your shoes with.

For example, you might know how to tie a “square knot” and a “granny knot.” They have the same steps except that one twist is reversed from the square knot to the granny knot. But can you prove that those knots are different? Well, knot theorists can.

✅ Up Next: The Amazing Math Inside the Rubik’s Cube

Knot theorists’ holy grail problem was an algorithm to identify if some tangled mess is truly knotted, or if it can be disentangled to nothing. The cool news is that this has been accomplished! Several computer algorithms for this have been written in the last 20 years, and some of them even animate the process .

But the Unknotting Problem remains computational. In technical terms, it’s known that the Unknotting Problem is in NP, while we don ’ t know if it’s in P. That roughly means that we know our algorithms are capable of unknotting knots of any complexity, but that as they get more complicated, it starts to take an impossibly long time. For now.

If someone comes up with an algorithm that can unknot any knot in what’s called polynomial time, that will put the Unknotting Problem fully to rest. On the flip side, someone could prove that isn’t possible, and that the Unknotting Problem’s computational intensity is unavoidably profound. Eventually, we’ll find out.

The Large Cardinal Project

If you’ve never heard of Large Cardinals , get ready to learn. In the late 19th century, a German mathematician named Georg Cantor figured out that infinity comes in different sizes. Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it.

There is the first infinite size, the smallest infinity , which gets denoted ℵ₀. That’s a Hebrew letter aleph; it reads as “aleph-zero.” It’s the size of the set of natural numbers, so that gets written |ℕ|=ℵ₀.

Next, some common sets are larger than size ℵ₀. The major example Cantor proved is that the set of real numbers is bigger, written |ℝ|>ℵ₀. But the reals aren’t that big; we’re just getting started on the infinite sizes.

✅ More Mind-Blowing Stuff: Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things

For the really big stuff, mathematicians keep discovering larger and larger sizes, or what we call Large Cardinals. It’s a process of pure math that goes like this: Someone says, “I thought of a definition for a cardinal, and I can prove this cardinal is bigger than all the known cardinals.” Then, if their proof is good, that’s the new largest known cardinal. Until someone else comes up with a larger one.

Throughout the 20th century, the frontier of known large cardinals was steadily pushed forward. There’s now even a beautiful wiki of known large cardinals , named in honor of Cantor. So, will this ever end? The answer is broadly yes, although it gets very complicated.

In some senses, the top of the large cardinal hierarchy is in sight. Some theorems have been proven, which impose a sort of ceiling on the possibilities for large cardinals. But many open questions remain, and new cardinals have been nailed down as recently as 2019. It’s very possible we will be discovering more for decades to come. Hopefully we’ll eventually have a comprehensive list of all large cardinals.

What’s the Deal with 𝜋+e?

Given everything we know about two of math’s most famous constants, 𝜋 and e , it’s a bit surprising how lost we are when they’re added together.

This mystery is all about algebraic real numbers . The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.

✅ Try It Yourself: Can You Solve This Viral Brain Teaser From TikTok?

All rational numbers, and roots of rational numbers, are algebraic. So it might feel like “most” real numbers are algebraic. Turns out, it’s actually the opposite. The antonym to algebraic is transcendental, and it turns out almost all real numbers are transcendental—for certain mathematical meanings of “almost all.” So who’s algebraic , and who’s transcendental?

The real number 𝜋 goes back to ancient math, while the number e has been around since the 17th century. You’ve probably heard of both, and you’d think we know the answer to every basic question to be asked about them, right?

Well, we do know that both 𝜋 and e are transcendental. But somehow it’s unknown whether 𝜋+e is algebraic or transcendental. Similarly, we don’t know about 𝜋e, 𝜋/e, and other simple combinations of them. So there are incredibly basic questions about numbers we’ve known for millennia that still remain mysterious.

Is 𝛾 Rational?

Here’s another problem that’s very easy to write, but hard to solve. All you need to recall is the definition of rational numbers.

Rational numbers can be written in the form p/q, where p and q are integers. So, 42 and -11/3 are rational, while 𝜋 and √2 are not. It’s a very basic property, so you’d think we can easily tell when a number is rational or not, right?

Meet the Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.

✅ One More Thing: Teens Have Proven the Pythagorean Theorem With Trigonometry. That Should Be Impossible

The sleek way of putting words to those symbols is “gamma is the limit of the difference of the harmonic series and the natural log.” So, it’s a combination of two very well-understood mathematical objects. It has other neat closed forms, and appears in hundreds of formulas.

But somehow, we don’t even know if 𝛾 is rational. We’ve calculated it to half a trillion digits, yet nobody can prove if it’s rational or not. The popular prediction is that 𝛾 is irrational. Along with our previous example 𝜋+e, we have another question of a simple property for a well-known number, and we can’t even answer it.

Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.

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What does is really mean to say that a 3-body problem is not solvable? [duplicate]

What does it really mean to say that a three-body problem (the Sun, the earth, and the moon) is not solvable? Why is it not possible to solve the differential equations on a computer with adequate initial conditions? What's the real issue here?

• newtonian-mechanics
• newtonian-gravity
• computational-physics
• differential-equations
• three-body-problem

• 1 $\begingroup$ Related: Why does the three-body problem have no solution? More on 3-body problem . $\endgroup$ –  Qmechanic ♦ Apr 12, 2022 at 14:13
• 3 $\begingroup$ TLDR, "solve" means, "find a closed form expression for the motion of the bodies." A numerical simulation of their motion doesn't count as a "solution." $\endgroup$ –  Solomon Slow Apr 12, 2022 at 15:06

4 Answers 4

The three-body problem lacks a closed-form solution, which is a mathematical expression that uses a finite number of "standard" operations (addition, division, logarithm, etc.), usually expressed as a formula or equation. This means there is no equation for which you can plug in the initial positions, velocities, and masses, and solve for the exact positions and velocities at a later time. For most configurations of the three bodies, numerical methods are needed to iteratively compute the positions over time, although the chaotic behavior of the system means that even small numerical errors can propagate and result in large deviations over time.

Chaotic systems are very sensitive to their input parameters, so anything less than a perfect continuous estimation can result in wildly different results. Unlike a non-chaotic problem where a 1% error in velocity might result in a 1% error in position, a 1% error in velocity at some time step can result in arbitrarily large deviation at a later time. Even if your numerical approximation is very good, you may eventually find that your model predicts something entirely different from reality. You can run increasingly good approximations with decreasingly short time steps, but there is no numerical approach that is anything but an approximation, which simply may not be "good enough" in a chaotic system.

Of course, the three body problem does have a solution - it's whatever would actually happen when observing the three bodies in isolation. The physical reality is the solution to the problem. It's just that this solution cannot be expressed with common mathematical operations in "closed form".

See Why does the three-body problem have no solution?

Quoting from wikipedia

The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

Of course, you can get a numerical solution for given initial conditions and masses. (Although, as Jon Custer points out, getting accurate numerical solutions is very difficult).

• 4 $\begingroup$ Now, whether you get the correct solution is another question, particularly since numerical methods have limited precision, and slightly different initial conditions lead to that chaos thing... $\endgroup$ –  Jon Custer Apr 12, 2022 at 13:43
• $\begingroup$ So the point is that numerical solutions exist, but no one has yet found an analytical solution? $\endgroup$ –  Solidification Apr 12, 2022 at 13:45
• $\begingroup$ @JonCuster Good point! $\endgroup$ –  Andrew Apr 12, 2022 at 13:47
• 1 $\begingroup$ @Solidification I would think that the "yet" is in the realm of "impossible". $\endgroup$ –  anna v Apr 12, 2022 at 13:49
• $\begingroup$ @Solidification Basically. Although (a) as Jon points out, the numerical solutions are very hard, and (b) it's not just that no one has found analytical solutions yet, it's that the solutions are known to have properties (chaos) that are extremely hard to represent analytically (like annav says, 'extremely hard' here probably means impossible to write a closed form expression in terms of simple functions). $\endgroup$ –  Andrew Apr 12, 2022 at 13:49

There has been always discussion about what is called solvable in the community.

For example: Take the pendulum with a constant force acting on it (and not applying the small angle approximation, otherwise the solution is a simple sine function), which is given by $$ {\ddot {x}} = -\sin(x) \,.$$ With the small angle approximation $$ {\ddot {x}} = -x\,,$$ you can solve this directly with the well-known function $$ x = \sin(t) + \phi_0\,.$$

However, you can solve the full (non-linear) equations of motion as a perturbation series. One could take the ansatz of writing $x$ as $$ x = \sum_{\nu=0}^N \alpha_\nu \frac{t^\nu}{\nu!}$$ To lowest order in time $t$ we would then have $$ \alpha_2+ \alpha_3 t +\mathcal{O}(t^2) = -\sin(\alpha_0+\alpha_1 t + \mathcal{O}(t^2) ) $$

which gives with the Taylor expansion of the sine $$ \alpha_2 +\mathcal{O}(t) = -\alpha_0+\mathcal{O}(t) $$ You can then solve this equation order by order for each $\alpha_\nu$ . This can be done up to any order you want. Some people would say that this is not an analytical solution, but keep in mind, that depending on your philosophy the sine function is just a name for some perturbation series that has a particular structure. Just because the perturbation series that solves the above differential equation does not have a name, does not make the solution any less important.

Now addressing the question: You can solve the differential equation describing the three body problem numerically, but people usually mean that there exists no well known function that describe the solution in easy terms. There is no 'sine' function for the three body problem. Note that this is different from the two body problem, where we know the solution of $r(\phi)$ to be quite simple.

Additionally, the three body problem is chaotic (as was pointed out by Andrew), so any small change in the initial conditions will lead to large deviations for predictions far in the future. Numerically, chaotic systems are not easy problems, since you do not want unphysical choices (like the time step you choose) to influence your final result, but this happens often in simulations of these systems.

• $\begingroup$ If I'm not wrong, the equation $\ddot{x}=-\sin x$ is exactly solvable (in terms of elliptic integrals). $\endgroup$ –  Solidification Apr 12, 2022 at 14:28
• 1 $\begingroup$ @Solidification afaik, special functions like those are not generally considered "closed form", wikipedia actually has a nice table of what's included in each category en.wikipedia.org/wiki/… $\endgroup$ –  llama Apr 13, 2022 at 5:01

It is Solvable...

Why is it not possible to solve the differential equations on a computer with adequate initial conditions?

It is possible. But you will be hard-pressed to build a computer which can represent the "adequate initial conditions". We should say that a problem $P(x)$ is "solvable", if, for any input $x$ , you give me a precise answer for $P$ .

In the case of the sun-earth-moon system, that means that for arbitrary starting conditions and an arbitrary point in the future, you can tell me the positions of each body. For any system with precise dynamical equations, you can just plug in the initial conditions and iterate to the desired endpoint. What closed-form solutions give you is the ability to jump directly to the endpoint without performing any of the intermediate calculations. Obviously, this is very powerful. But it's more than that. It may be the only way to obtain a correct solution at all.

...In the Integers

The problem with a computer ( all computers, including you ) is that it can only compute with integers. IEEE 754 makes it look like you can compute with reals, but you are not. Floating-point numbers are just integers dressed up to look like reals, but actually doing a very poor job of it. You can perform exact symbolic operations on reals in the manner of Mathematica and friends, but the set of reals which can be handled in this manner is infinitely smaller than the full set of reals (due to the fact that all such reals must have a finite-length symbol attached to them, which just puts them in correspondence with the integers). Which means, the operations you can perform may quickly get exhausted while trying to obtain the answer you are looking for.

If your universe is quantized, and thus can be exactly described by integers, then, in principle, all 3-body problems should be exactly solvable. If time or space cannot be exactly represented by integers, then any computations you perform on them will contain errors. And any sequence of computations will accumulate growing errors.

Exponential Error

Some problems are sufficiently simple that the errors tend to cancel out, and one can feasibly obtain answers of arbitrary precision by performing computations of arbitrary precision (the precision of the result is proportional to the precision of the inputs). Simple harmonic motions, two-body problems, etc. fall into this category. When this happens, there is usually a closed-form solution which lets you jump straight to the desired endpoint.

When a physics problem is "not solvable", it generally means that despite an exactly precise dynamical description of the system (so far as we can tell, anyway), we cannot compute future states with arbitrary precision, because the accumulated error may grow without bound (i.e., "chaotic" systems). You can always increase the precision of the result by increasing the precision of the inputs and intermediate calculations; but if the error grows exponentially, then it quickly becomes infeasible to meaningfully improve the precision of the result by increasing the precision upstream of it. And this is what I mean by having trouble with "adequate initial conditions". If you want 64 bits of precision in your result, you might need something like 2^64 bits of precision in the initial conditions and intermediate calculations to achieve that. You can't build a computer big enough to get that answer.

That being said, some 3-body systems have "nice properties" which make their behavior not too chaotic. The bodies in the sun-moon-earth system are of masses and distances that we can compute useful ephemerides over most of the time periods of interest. Whereas, trying to determine which comets might collide with a planet in the next billion years might not be feasible at all, even given arbitrarily precise orbital elements.

Not the answer you're looking for? Browse other questions tagged newtonian-mechanics newtonian-gravity computational-physics differential-equations three-body-problem or ask your own question .

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• » 10 Math Equations That Have Never Been Solved

10 Math Equations That Have Never Been Solved

By Kathleen Cantor, 10 Sep 2020

Mathematics has played a major role in so many life-altering inventions and theories. But there are still some math equations that have managed to elude even the greatest minds, like Einstein and Hawkins. Other equations, however, are simply too large to compute. So for whatever reason, these puzzling problems have never been solved. But what are they?

Like the rest of us, you're probably expecting some next-level difficulty in these mathematical problems. Surprisingly, that is not the case. Some of these equations are even based on elementary school concepts and are easily understandable - just unsolvable.

1. The Riemann Hypothesis

Equation: σ (n) ≤ Hn +ln (Hn)eHn

• Where n is a positive integer
• Hn is the n-th harmonic number
• σ(n) is the sum of the positive integers divisible by n

For an instance, if n = 4 then σ(4)=1+2+4=7 and H4 = 1+1/2+1/3+1/4. Solve this equation to either prove or disprove the following inequality n≥1? Does it hold for all n≥1?

This problem is referred to as Lagarias’s Elementary Version of the Riemann Hypothesis and has a price of a million dollars offered by the  Clay Mathematics Foundation  for its solution.

2. The Collatz Conjecture

Equation: 3n+1

• where n is a positive integer n/2
• where n is a non-negative integer

Prove the answer end by cycling through 1,4,2,1,4,2,1,… if n is a positive integer. This is a repetitive process and you will repeat it with the new value of n you get. If your first n = 1 then your subsequent answers will be 1, 4, 2, 1, 4, 2, 1, 4… infinitely. And if n = 5 the answers will be 5,16,8,4,2,1 the rest will be another loop of the values 1, 4, and 2.

This equation was formed in 1937 by a man named Lothar Collatz which is why it is referred to as the Collatz Conjecture.

3. The Erdős-Strauss Conjecture

Equation: 4/n=1/a+1/b+1/c

• a, b and c are positive integers.

This equation aims to see if we can prove that for if n is greater than or equal to 2, then one can write 4*n as a sum of three positive unit fractions.

This equation was formed in 1948 by two men named Paul Erdős and Ernst Strauss which is why it is referred to as the Erdős-Strauss Conjecture.

4. Equation Four

Equation: Use 2(2∧127)-1 – 1 to prove or disprove if it’s a prime number or not?

Looks pretty straight forward, does it? Here is a little context on the problem.

Let’s take a prime number 2. Now, 22 – 1 = 3 which is also a prime number. 25 – 1 = 31 which is also a prime number and so is 27−1=127. 2127 −1=170141183460469231731687303715884105727 is also prime.

5. Goldbach's Conjecture

Equation: Prove that x + y = n

• where x and y are any two primes

This problem, as relatively simple as it sounds has never been solved. Solving this problem will earn you a free million dollars. This equation was first proposed by Goldbach hence the name Goldbach's Conjecture.

If you are still unsure then pick any even number like 6, it can also be expressed as 1 + 5, which is two primes. The same goes for 10 and 26.

6. Equation Six

Equation: Prove that (K)n = JK1N(q)JO1N(q)

• Where O = unknot (we are dealing with  knot theory )
• (K)n  =  Kashaev's invariant of K for any K or knot
• JK1N(q) of K is equal to N- colored Jones polynomial
• We also have the volume of conjecture as (EQ3)
• Here vol(K)  =  hyperbolic volume

This equation tries to portray the relationship between  quantum invariants  of knots and  the hyperbolic geometry  of  knot complements . Although this equation is in mathematics, you have to be a physics familiar to grasp the concept.

7. The Whitehead Conjecture

Equation: G = (S | R)

• when CW complex K (S | R) is aspherical
• if π2 (K (S | R)) = 0

What you are doing in this equation is prove the claim made by Mr.  Whitehead  in 1941 in  an algebraic topology  that every subcomplex of an  aspherical   CW complex  that is connected and in two dimensions is also spherical. This was named after the man, Whitehead conjecture.

8. Equation Eight

Equation: (EQ4)

• Where Γ = a  second countable   locally compact group
• And the * and r subscript = 0 or 1.

This equation is the definition of  morphism  and is referred to as an assembly map.  Check out the  reduced C*-algebra  for more insight into the concept surrounding this equation.

9. The Euler-Mascheroni Constant

Equation: y=limn→∞(∑m=1n1m−log(n))

Find out if y is rational or irrational in the equation above. To fully understand this problem you need to take another look at rational numbers and their concepts.  The character y is what is known as the Euler-Mascheroni constant and it has a value of 0.5772.

This equation has been calculated up to almost half of a trillion digits and yet no one has been able to tell if it is a rational number or not.

10. Equation Ten

Equation: π + e

Find the sum and determine if it is algebraic or transcendental. To understand this question you need to have an idea of  algebraic real numbers  and how they operate. The number pi or π originated in the 17th century and it is transcendental along with e. but what about their sum? So Far this has never been solved.

As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you're looking for a brain teaser, finding the solutions to these problems will give you a run for your money.

See the 11 Comments below.

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Posted in Mathematics category - 10 Sep 2020 [ Permalink ]

11 Comments on “10 Math Equations That Have Never Been Solved”

But 2(2127)−1 = 340282366920938463463374607431768211455 is not a prime number. It is divisible by 64511.

Hello I am explorer and i type on google search " unsolvable mathematical formulas ", and I first find this syte. I see you are good-math-guys. Do you know what is this formula means:

π × ∞ = " 5 "

If you happen to have a quantum computer, I am not kidding be smart and don't insert this formula: [π × ∞ = " 5 "] into it please.

Maybe only, if you know meaning of this three symbols up writen and connected together.

(x dot epsilon)

I can explain my theory if you want me to spoil the pleasure of solving the equation. And mathematics as a science too or " as well " sorry i am not good in English, and google translate is not exelent.

8.539728478 is the answer to number 10

8.539728478 is the answer to number 10 or 8.539734221

Equation Four: Solved

To determine whether the number 2(2^127)-1 – 1 is a prime number, we first need to calculate its value. The expression 2(2^127) can be simplified as follows:

2(2^127) = 2 * 2^127 = 2^128

Therefore, the expression 2(2^127)-1 – 1 can be written as 2^128 – 1 – 1. We can then simplify this further to get:

2^128 – 1 – 1 = 2^128 – 2

To determine whether this number is prime, we can use the fundamental theorem of arithmetic, which states that every positive integer can be written as a product of prime numbers in a unique way (ignoring the order of the factors). This means that if a number is not prime, it can be expressed as the product of two or more prime numbers.

We can use this theorem to determine whether 2^128 – 2 is prime by trying to express it as the product of two or more prime numbers. However, it is not possible to do this, because 2^128 – 2 cannot be evenly divided by any prime number (except for 1, which is not considered a prime number).

Therefore, we can conclude that 2^128 – 2 is a prime number, because it cannot be expressed as the product of two or more prime numbers.

Equation Ten: Solved

The sum of π and e is equal to π + e = 3.14159 + 2.71828 = 5.85987.

To determine whether this number is algebraic or transcendental, we first need to understand the difference between these two types of numbers. Algebraic numbers are numbers that can be expressed as a root of a polynomial equation with integer coefficients, while transcendental numbers cannot be expressed in this way.

In this case, the number 5.85987 can be expressed as the root of the polynomial equation x^2 - 5.85987x + 2.71828 = 0. Therefore, it is an algebraic number.

In conclusion, the sum of π and e is equal to 5.85987, which is an algebraic number.

Equation 2: SOLVED

The equation 3n + 1 states that a positive integer n should be multiplied by 3 and then 1 should be added to the result. If the resulting value is then divided by 2 and the quotient is a non-negative integer, the process should be repeated with the new value of n.

To prove that this equation always results in a repeating sequence of 1, 4, 2, 1, 4, 2, 1, ... if n is a positive integer, we can start by substituting a value for n and performing the calculations as specified in the equation.

For example, if n is equal to 1, the sequence of values will be: n = 1 3n + 1 = 3(1) + 1 = 4 n = 4/2 = 2 3n + 1 = 3(2) + 1 = 7 n = 7/2 = 3.5

Since n must be a non-negative integer, the value of n becomes 2 when the result of the previous calculation is divided by 2. The sequence then becomes: n = 2 3n + 1 = 3(2) + 1 = 7 n = 7/2 = 3.5

The value of n becomes 3 when the result of the previous calculation is divided by 2. The sequence then becomes: n = 3 3n + 1 = 3(3) + 1 = 10 n = 10/2 = 5

The value of n becomes 5 when the result of the previous calculation is divided by 2. The sequence then becomes:

n = 5 3n + 1 = 3(5) + 1 = 16 n = 16/2 = 8 n = 8/2 = 4 n = 4/2 = 2 n = 2/2 = 1 n = 1/2 = 0.5

Since n must be a non-negative integer, the value of n becomes 1 when the result of the previous calculation is divided by 2. The sequence then becomes: n = 1 3n + 1 = 3(1) + 1 = 4 n = 4/2

To complete the proof that the equation 3n + 1 always results in a repeating sequence of 1, 4, 2, 1, 4, 2, 1, ... if n is a positive integer, we can continue the process of substituting values for n and performing the calculations as specified in the equation.

If n is equal to 4, the sequence of values will be: n = 4 3n + 1 = 3(4) + 1 = 13 n = 13/2 = 6.5

Since n must be a non-negative integer, the value of n becomes 6 when the result of the previous calculation is divided by 2. The sequence then becomes:

n = 6 3n + 1 = 3(6) + 1 = 19 n = 19/2 = 9.5

Certainly! To complete the proof that the equation 3n + 1 always results in a repeating sequence of 1, 4, 2, 1, 4, 2, 1, ... if n is a positive integer, we can continue the process of substituting values for n and performing the calculations as specified in the equation.

If n is equal to 4, the sequence of values will be:

n = 4 3n + 1 = 3(4) + 1 = 13 n = 13/2 = 6.5

Since n must be a non-negative integer, the value of n becomes 9 when the result of the previous calculation is divided by 2. The sequence then becomes:

n = 9 3n + 1 = 3(9) + 1 = 28 n = 28/2 = 14 n = 14/2 = 7 n = 7/2 = 3.5

The value of n becomes 3 when the result of the previous calculation is divided by 2. The sequence then becomes:

n = 3 3n + 1 = 3(3) + 1 = 10 n = 10/2 = 5 n = 5/2 = 2.5

Since n must be a non-negative integer, the value of n becomes 2 when the result of the previous calculation is divided by 2. The sequence then becomes:

n = 2 3n + 1 = 3(2) + 1 = 7 n = 7/2 = 3.5

As we can see, the sequence of values becomes repetitive

The Riemann Hypothesis

This equation states that the sum of the positive integers divisible by n (σ(n)) is less than or equal to the n-th harmonic number (Hn) plus the natural logarithm of the n-th harmonic number (ln(Hn)) multiplied by the n-th harmonic number (Hn) raised to the power of Hn.

To solve this equation, you would need to substitute a specific value for n and determine the value of Hn and σ(n) for that specific value. You can then substitute these values into the equation and see if it holds true.

For example, if n = 5, the sum of the positive integers divisible by 5 (σ(5)) is 15 (1 + 5 + 10 + 15 + 20 + 25), the 5th harmonic number (H5) is 2.28, and the natural logarithm of the 5th harmonic number (ln(H5)) is 0.83. Substituting these values into the equation, we get:

σ(5) ≤ H5 + ln(H5)eH5 15 ≤ 2.28 + 0.83 * 2.28^2.28 15 ≤ 4.39

Since 15 is less than or equal to 4.39, the equation holds true for this specific value of n.

Equation #9

In the equation y = limn→∞(∑m=1n1m−log(n)), y is the limit of the sequence (∑m=1n1m−log(n)) as n approaches infinity.

The Euler-Mascheroni constant is defined as the limit of the sequence (∑m=1n1m−log(n)) as n approaches infinity, and it has a value of approximately 0.5772. Therefore, y is equal to the Euler-Mascheroni constant, which is a rational number.

Rational numbers are numbers that can be expressed as the ratio of two integers, such as 3/4, 7/11, or 2/5. They can be written as a finite or repeating decimal, such as 0.75, 0.636363636..., or 1.5.

Irrational numbers are numbers that cannot be expressed as the ratio of two integers, and they cannot be written as a finite or repeating decimal. Examples of irrational numbers include √2, π, and e.

Since y is equal to the Euler-Mascheroni constant, which is a rational number, y is a rational number.

The equation G = (S | R) is a definition of a CW complex, where S and R are subcomplexes of G. A CW complex is a topological space that can be built up from cells, where each cell is homeomorphic to a closed ball in Euclidean space.

The statement "when CW complex K (S | R) is aspherical" means that the complex K (S | R) does not contain any non-trivial loops, i.e. loops that cannot be continuously contracted to a point. This implies that the fundamental group of K (S | R) is trivial, which means that π1(K (S | R)) = {e}.

The statement "if π2 (K (S | R)) = 0" means that the second homotopy group of the complex K (S | R) is trivial, which means that there are no non-trivial 2-dimensional holes in K (S | R).

Together, these statements imply that the CW complex K (S | R) is a topological space with no non-trivial loops or holes. This is a strong condition that is satisfied by very few spaces, and it is a necessary condition for a space to be aspherical.

In summary, the statement "when CW complex K (S | R) is aspherical" and "if π2 (K (S | R)) = 0" implies that the complex K (S | R) is a topological space with no non-trivial loops or holes, which is a necessary condition for a space to be aspherical.

#3 Erdos Strauss Conjecture:

To solve the equation 4/n = 1/a + 1/b + 1/c where n ≥ 2, a, b and c are positive integers, we can first multiply both sides of the equation by nabc to get rid of the fractions:

4abc = nab + nbc + nac

We can then group like terms:

4abc = (n + a)(b + c)

Now we can use the fact that n, a, b, and c are positive integers to make some observations:

Since n, a, b and c are positive integers, n, a, b and c must be factors of 4abc. Since n is greater than or equal to 2, it must be one of the factors of 4abc. The other factors of 4abc are (n + a), b, and c. So, to find all the possible values of n, a, b, and c, we must find all the ways to factorize 4abc such that one of the factors is greater than or equal to 2.

4abc = 4 * 1 * 1 * 2 * 3 * 5 = 120

Some possible factorizations are:

n = 2, a = 1, b = 5, c = 12 n = 2, a = 3, b = 5, c = 8 n = 2, a = 4, b = 3, c = 15 n = 2, a = 6, b = 2, c = 20 n = 4, a = 1, b = 3, c = 30 So, the possible solutions to the equation are: (n,a,b,c) = (2,1,5,12), (2,3,5,8), (2,4,3,15), (2,6,2,20), (4,1,3,30)

It's worth noting that this is not an exhaustive list, but just some of the possible solutions, as there could be infinitely many solutions to this equation.

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Biden, McCarthy appear near two-year deal on US debt ceiling as default looms

WASHINGTON, May 25 (Reuters) - U.S. President Joe Biden and top congressional Republican Kevin McCarthy are closing in on a deal that would raise the government's $31.4 trillion debt ceiling for two years while capping spending on most items, a U.S. official told Reuters.

The deal, which is not final, would increase funding for discretionary spending on military and veterans while essentially holding non-defense discretionary spending at current year levels, the official said, who requested anonymity because they are not authorized to speak about internal discussions.

The White House is considering scaling back its plan to boost funding at the Internal Revenue Service to hire more auditors and target wealthy Americans, the official said.

A second U.S. official said IRS funding is an open issue, but the main thrust is ensuring the agency executes the president's priorities, even if there is a small haircut or funding is moved around.

The final deal would specify the total amount the government could spend on discretionary programs like housing and education, according to a person familiar with the talks, but not break that down into individual categories. The two sides are just $70 billion apart on a total figure that would be well over $1 trillion, according to another source.

The two sides met virtually on Thursday, the White House said.

Republican negotiators have backed off plans to increase military spending while cutting non-defense spending and instead backed a White House push to treat both budget items more equally, a source familiar with the talks told Reuters.

Biden said they still disagreed over where the cuts should fall.

"I don't believe the whole burden should fall back to middle class and working-class Americans," he told reporters.

House Speaker McCarthy told reporters Thursday evening the two sides have not reached a deal. "We knew this would not be easy," he said.

It is unclear precisely how much time Congress has left to act. The Treasury Department was warned that it could be unable to cover all its obligations as soon as June 1, but on Thursday said it would sell $119 billion worth of debt that will come due on that date, suggesting to some market watchers that it was not an iron-clad deadline.

"They have suggested in the past that they would not announce auctions that they did not believe they had the means to settle," Gennadiy Goldberg, senior rates strategist at TD Securities in New York. "So I do think that's a positive note."

Any agreement will have to pass the Republican-controlled House of Representatives and the Democratic-controlled Senate. That could be tricky, as some right-wing Republicans and many liberal Democrats said they were upset by the prospect of compromise.

"I don't think everybody's going to be happy at the end of the day. That's not how the system works," McCarthy said.

The House adjourned on Thursday afternoon for a week-long break, and the Senate is not in session. Lawmakers have been told to be ready to come back to vote if a deal is reached.

The deal would only set broad spending outlines, leaving lawmakers to fill in the blanks in the weeks and months to come.

Biden has resisted Republican proposals to stiffen work requirements for anti-poverty programs and loosen oil and gas drilling rules, according to Democratic Representative Mark Takano.

Representative Kevin Hern, who leads the powerful Republican Study Committee, told Reuters a deal was likely by Friday afternoon.

'TIME'S UP'

Democrats on Thursday focused their attacks on what they said would be devastating cuts in federal aid for veterans - ranging from healthcare and food aid to housing assistance - if Republicans got their way in the negotiations.

"Time's up for all of these games around here," Democratic Representative Don Davis, a U.S. Air Force veteran, said at a press conference.

A U.S. default could upend global financial markets and push the United States into recession.

Credit rating agency DBRS Morningstar put the United States on review for a possible downgrade on Thursday, echoing similar warnings by Fitch , Moody's and Scope Ratings. Another agency, S&P Global, downgraded U.S. debt following a similar debt-ceiling standoff in 2011.

The months-long standoff has spooked Wall Street, weighing on U.S. stocks and pushing the nation's cost of borrowing higher.

Deputy Treasury Secretary Wally Adeyemo said concerns about the debt ceiling had pushed up the government's interest costs by $80 million so far.

Lawmakers regularly need to raise the self-imposed debt limit to cover the cost of spending and tax cuts they have already approved.

House lawmakers will get three days to read any debt-ceiling bill before they have to vote on it.

McCarthy has insisted that any deal must cut discretionary spending next year and cap spending growth in the years to come, to slow the growth of the U.S. debt, now equal to the annual output of the economy.

Lawmakers on the parties' right and left flanks are growing frustrated. Republican Representative Chip Roy, a member of the hard-right Freedom Caucus, has insisted that any deal must include the sharp spending cuts they passed last month.

Some Democrats, meanwhile, say Biden has not been vocal enough about the downsides to Republicans' proposed spending cuts, in contrast to McCarthy who has been briefing reporters multiple times per day.

"I would urge the president to use the power of the bully pulpit of the presidency," said Democratic Representative Steven Horsford.

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