Math problems that are impossible to solve [closed]

Solution 1:

The proof that demonstrates the impossibility of trisecting an angle uses Galois theory. Galois theory can also be used to show that certain polygons cannot be constructed with compass and straightedge, and was originally used to show that, in general, polynomials of degree $\geq 5$ are not solvable.

To be specific, an $n$-gon is constructible via compass and straightedge $\displaystyle \iff n = 2^k \prod_{r=1}^m p_r$, for $k, m \in \mathbb{Z}_{\geq 0}$ and the $p_r$'s distinct Fermat primes.

Without getting into too much detail, Galois theory is a subset of abstract algebra which links together concepts in group theory and field theory, beginning with the observation that the set of automorphisms of a field forms a group. At any rate, I'm unsure of your level of background, so I'll just post a few Wikipedia links for to wet your appetite if you're interested:

• Galois theory
• Groups
• Fields
• Automorphisms

Of course, there are many, many answers to your question, so I'll leave my post at that and let others have the opportunity to post more.

Solution 2:

It is impossible to find a rational number whose square is 2.

Solution 3:

This answer takes a bit of background, but I think it's worth it. Please bear with me! If you already have the necessary background, you can skip to the last section for the punch line.

The surface of a sweet bun and the surface of a doughnut are both examples of two-dimensional manifolds: geometric spaces that look like planes from close up. An ant sitting on a sweet bun and an ant sitting on a doughnut see the same thing: a vast expanse of glazed pastry stretching off into the distance.

Although they don't have very good eyes, ants do have the ability to leave complicated scent trails as they walk around. Using this ability, even the tiniest ant can tell the difference between a sweet bun and a doughnut. One of the simplest ways to do this is to lay down trails that divide the surface into polygons, count the vertices, edges, and faces, and compute the Euler characteristic. With this technique, the ant can identify not only a sweet bun or a doughnut, but any pastry with a two-dimensional surface. (To make sure the ant doesn't have to walk forever, let's assume the pastry is compact—that is, finite in extent. If the pastry is infinite in extent, the ant should probably conclude that she's died and gone to ant heaven.)

To put it more technically, there's an algorithm an ant can use to identify any compact two-dimensional manifold.

There are also manifolds of other dimensions. A three-dimensional manifold, for example, is a geometric space that looks like a three-dimensional Euclidean space from close up. Our spacial universe is a classic example of a three-dimensional manifold. (In our best current cosmological models, it's the most boring one: just three-dimensional Euclidean space itself. However, it didn't have to be that way. If the density of the universe were a little bit higher, space would be curled up into the three-dimensional version of a sphere. Mark Peterson has argued, surprisingly convincingly, that the universe described in Dante's Divine Comedy has exactly this shape.)

You might guess that a clever ant should be able to identify a compact manifold of any dimension by laying down scent trails, maybe using a more sophisticated version of the technique we saw for two-dimensional manifolds. But you'd be wrong!

In 1960, Andrey Markov Jr. proved that for $n \ge 4$, there's no algorithm an ant can use to identify any compact $n$-dimensional manifold.

The reason is that an ant who could identify any compact four-dimensional manifold would also be able to identify any finitely presented group, and an ant who could do that would be able to solve the halting problem! James van Meter has a wonderful exposition of the proof, although it unavoidably requires some knowledge of abstract algebra and algebraic topology.

We've seen that compact two-dimensional manifolds can be identified algorithmically, but compact manifolds of dimension four and higher can't be. What about compact three-dimensional manifolds? As far as I can tell, we don't know! Our ant friend can certainly identify some of them, if she's willing to lug around this book, but I've never heard of an algorithm that works in general. If someone who knows more about low-dimensional topology could weigh in on this, I'd really appreciate it.

Solution 4:

Constructing an algorithm to solve any Diophantine equation has been proven to be equivalent to solving the halting problem, as is computing the Kolmogorov complexity (optimal compression size) of any given input, for any given universal description language.

In general I think what you're looking for is either problems that are proven to be equivalent to the halting problem, or problems that are formally independent from, say, ZFC, such as the Continuum Hypothesis or more generally Gödel's incompleteness theorems.

Solution 5:

It's impossible to define mathematics in a way that satisfies all mathematicians.

This is likely not the answer you're after, but it stroke me when I first heard of it because definitions in mathematics are both exact and omnipresent, and I was not suspecting that providing a universal definition of mathematics itself could be such an intractable problem.