In 2019, mathematicians finally solved one of the hardest math problems—one that had stumped them for decades. It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.

On the surface, it seems easy. Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42?

That turned out to be much harder—as in, no one was able to solve for those integers for 65 years until a supercomputer finally came up with the solution to 42. (For the record: x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Obviously.)

That’s the beauty of math: There’s always an answer for everything, even if takes years, decades, or even centuries to find it. So here are nine more brutally difficult math problems that once seemed impossible, until mathematicians found a breakthrough.

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1

## The Poincaré Conjecture

In 2000, the Clay Mathematics Institute, a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered $1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

Henri Poincaré was a French mathematician who, around the turn of the 20th century, did foundational work in what we now call topology. Here’s the idea: Topologists want mathematical tools for distinguishing abstract shapes. For shapes in 3D space, like a ball or a donut, it wasn’t very hard to classify them all. In some significant sense, a ball is the simplest of these shapes.

Poincaré then went up to 4-dimensional stuff, and asked an equivalent question. After some revisions and developments, the conjecture took the form of “Every simply-connected, closed 3-manifold is homeomorphic to S^3,” which essentially says “the simplest 4D shape is the 4D equivalent of a sphere.”

Still with us?

A century later, in 2003, a Russian mathematician named Grigori Perelman posted a proof of Poincaré’s conjecture on the modern open math forum *arXiv.* Perelman’s proof had some small gaps, and drew directly from research by American mathematician Richard Hamilton. It was groundbreaking, yet modest.

After the math world spent a few years verifying the details of Perelman’s work, the awards began. Perelman was offered the million-dollar Millennium Prize, as well as the Fields Medal, often called the Nobel Prize of Math. Perelman rejected both. He said his work was for the benefit of mathematics, not personal gain, and also that Hamilton, who laid the foundations for his proof, was at least as deserving of the prizes.

2

## Fermat’s Last Theorem

Pierre de Fermat was a 17th-century French lawyer and mathematician. Math was apparently more of a hobby for Fermat, and so one of history’s greatest math minds communicated many of his theorems through casual correspondence. He made claims without proving them, leaving them to be proven by other mathematicians decades, or even centuries, later. The most challenging of these has become known as Fermat’s Last Theorem.

It’s a simple one to write. There are many trios of integers (x,y,z) that satisfy x²+y²=z². These are known as the Pythagorean Triples, like (3,4,5) and (5,12,13). Now, do any trios (x,y,z) satisfy x³+y³=z³? The answer is no, and that’s Fermat’s Last Theorem.

Fermat famously wrote the Last Theorem by hand in the margin of a textbook, along with the comment that he had a proof, but could not fit it in the margin. For centuries, the math world has been left wondering if Fermat *really* had a valid proof in mind.

Flash forward 330 years after Fermat’s death to 1995, when British mathematician Sir Andrew Wiles finally cracked one of history’s oldest open problems. For his efforts, Wiles was knighted by Queen Elizabeth II and was awarded a unique honorary plaque in lieu of the Fields Medal, since he was just above the official age cutoff to receive a Fields Medal.

Wiles managed to combine new research in very different branches of math in order to solve Fermat’s classic number theory question. One of these topics, Elliptic Curves, was completely undiscovered in Fermat’s time, leading many to believe Fermat never really had a proof of his Last Theorem.

3

## The Classification of Finite Simple Groups

From solving Rubik’s Cube to proving a fact about body-swapping on *Futurama*, abstract algebra has a wide range of applications. Algebraic groups are sets that follow a few basic properties, like having an “identity element,” which works like adding 0.

Groups can be finite or infinite, and if you want to know what groups of a particular size *n* look like, it can get very complicated depending on your choice of *n*.

If *n* is 2 or 3, there’s only one way that group can look. When *n* hits 4, there are two possibilities. Naturally, mathematicians wanted a comprehensive list of all possible groups for any given size.

The complete list took decades to finish conclusively, because of the difficulties in being sure that it was indeed complete. It’s one thing to describe what infinitely many groups look like, but it’s even harder to be sure the list covers everything. Arguably the greatest mathematical project of the 20th century, the classification of finite simple groups was orchestrated by Harvard mathematician Daniel Gorenstein, who in 1972 laid out the immensely complicated plan.

By 1985, the work was nearly done, but spanned so many pages and publications that it was unthinkable for one person to peer review. Part by part, the many facets of the proof were eventually checked and the completeness of the classification was confirmed.

By the 1990s, the proof was widely accepted. Subsequent efforts were made to streamline the titanic proof to more manageable levels, and that project is still ongoing today.

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4

## The Four Color Theorem

This one is as easy to state as it is hard to prove.

Grab any map and four crayons. It’s possible to color each state (or country) on the map, following one rule: No states that share a border get the same color.

The fact that any map can be colored with five colors—the Five Color Theorem—was proven in the 19th century. But getting that down to four took until 1976.

Two mathematicians at the University of Illinois, Urbana-Champaign, Kenneth Appel and Wolfgang Hakan, found a way to reduce the proof to a large, finite number of cases. With computer assistance, they exhaustively checked the nearly 2,000 cases, and ended up with an unprecedented style of proof.

Arguably controversial since it was partially conceived in the mind of a machine, Appel and Hakan’s proof was eventually accepted by most mathematicians. It has since become far more common for proofs to have computer-verified parts, but Appel and Hakan blazed the trail.

5

## (The Independence of) The Continuum Hypothesis

In the late 19th century, a German mathematician named Georg Cantor blew everyone’s minds by figuring out that infinities come in different sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors tend to learn in their Discrete Math classes.

Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the first infinite size; no infinite set is smaller than ℕ.

Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH).

If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.

So what’s the answer? This is where things take a turn.

CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.

It’s a weird state of affairs, but not completely uncommon in modern math. You may have heard of the Axiom of Choice, another independent statement. The proof of this outcome spanned decades and, naturally, split into two major parts: the proof that CH is consistent, and the proof that the negation of CH is consistent.

The first half is thanks to Kurt Gödel, the legendary Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel’s Constructible Universe, proved CH compatible with the baseline axioms, and is still a cornerstone of Set Theory classes. The second half was pursued for two more decades until Paul Cohen, a mathematician at Stanford, solved it by inventing an entire method of proof in Model Theory known as “forcing.”

Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles.

6

## Gödel’s Incompleteness Theorems

Gödel’s work in mathematical logic was totally next-level. On top of proving stuff, Gödel also liked to prove whether or not it was possible to *prove stuff*. His Incompleteness Theorems are often misunderstood, so here’s a perfect chance to clarify them.

Gödel’s First Incompleteness Theorem says that, in any proof language, there are always unprovable statements. There’s always something that’s true, that you can’t prove true. It’s possible to understand a (non-mathematically rigorous) version of Gödel’s argument, with some careful thinking. So buckle up, here it is: Consider the statement, “This statement cannot be proven true.”

Think through every case to see why this is an example of a true, but unprovable statement. If it’s false, then what it says is false, so then it can be proven true, which is contradictory, so this case is impossible. On the other extreme, if it did have a proof, then that proof would prove it true … making it true that it has no proof, which is contradictory, killing this case. So we’re logically left with the case that the statement is true, but has no proof. Yeah, our heads are spinning, too.

But follow that nearly-but-not-quite-paradoxical trick, and you’ve illustrated that Gödel’s First Incompleteness Theorem holds.

Gödel’s Second Incompleteness Theorem is similarly weird. It says that mathematical “formal systems” can’t prove themselves consistent. A consistent system is one that won’t give you any logical contradictions.

Here’s how you can think of that. Imagine Amanda and Bob each have a set of mathematical axioms—baseline math rules—in mind. If Amanda can use her axioms to prove that Bob’s axiom system is free of contradictions, then it’s impossible for Bob to use his axioms to prove Amanda’s system doesn’t yield contradictions.

So when mathematicians debate the best choices for the essential axioms of mathematics (it’s much more common than you might imagine) it’s crucial to be aware of this phenomenon.

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7

## The Prime Number Theorem

There are plenty of theorems about prime numbers. One of the simplest facts—that there are infinitely many prime numbers—can even be adorably fit into haiku form.

The Prime Number Theorem is more subtle; it describes the distribution of prime numbers along the number line. More precisely, it says that, given a natural number N, the number of primes below N is approximately N/log(N) ... with the usual statistical subtleties to the word “approximately” there.

Drawing on mid-19th-century ideas, two mathematicians, Jacques Hadamard and Charles Jean de la Vallée Poussin, independently proved the Prime Number Theorem in 1898. Since then, the proof has been a popular target for rewrites, enjoying many cosmetic revisions and simplifications. But the impact of the theorem has only grown.

The usefulness of the Prime Number Theorem is huge. Modern computer programs that deal with prime numbers rely on it. It’s fundamental to primality testing methods, and all the cryptology that goes with that.

8

## Solving Polynomials by Radicals

Remember the quadratic formula? Given ax²+bx+c=0, the solution is x=(-b±√(b^2-4ac))/(2a), which may have felt arduous to memorize in high school, but you have to admit is a conveniently closed-form solution.

Now, if we go up to ax³+bx²+cx+d=0, a closed form for “x=” is possible to find, although it’s much bulkier than the quadratic version. It’s also possible, yet ugly, to do this for degree 4 polynomials ax⁴+bx³+cx²+dx+f=0.

The goal of doing this for polynomials of any degree was noted as early as the 15th century. But from degree 5 on, a closed form is not possible. Writing the forms when they’re possible is one thing, but how did mathematicians prove it’s not possible from 5 up?

The world was only starting to comprehend the brilliance of French mathematician Evariste Galois when he died at the age of 20 in 1832. His life included months spent in prison, where he was punished for his political activism, writing ingenious, yet unrefined mathematics to scholars, and it ended in a fatal duel.

Galois’ ideas took decades after his death to be fully understood, but eventually they developed into an entire theory now called Galois Theory. A major theorem in this theory gives exact conditions for when a polynomial can be “solved by radicals,” meaning it has a closed form like the quadratic formula. All polynomials up to degree 4 satisfy these conditions, but starting at degree 5, some don’t, and so there’s no general form for a solution for any degree higher than 4.

9

## Trisecting an Angle

To finish, let’s go way back in history.

The Ancient Greeks wondered about constructing lines and shapes in various ratios, using the tools of an unmarked compass and straightedge. If someone draws an angle on some paper in front of you, and gives you an unmarked ruler, a basic compass, and a pen, it’s possible for you to draw the line that cuts that angle exactly in half. It’s a quick four steps, nicely illustrated like this, and the Greeks knew it two millennia ago.

What eluded them was cutting an angle in thirds. It stayed elusive for literally 15 centuries, with hundreds of attempts in vain to find a construction. It turns out such a construction is impossible.

Modern math students learn the angle trisection problem—and how to prove it’s not possible—in their Galois Theory classes. But, given the aforementioned period of time it took the math world to process Galois’ work, the first proof of the problem was due to another French mathematician, Pierre Wantzel. He published his work in 1837, 16 years after the death of Galois, but nine years before most of Galois’ work was published.

Either way, their insights are similar, casting the construction question into one about properties of certain representative polynomials. Many other ancient construction questions became approachable with these methods, closing off some of the oldest open math questions in history.

So if you ever time-travel to ancient Greece, you can tell them their attempts at the angle trisection problem are futile.

Dave Linkletter

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.