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Problem: Does every triangular billiard have a periodic orbit? For acute triangles, the question has been answered affirmatively by Fagnano in 1775: one can simply take the length $3$ orbit joining the basepoints of the heights of the triangle. For (generic) obtuse triangles, the answer is not known in spite of very considerable efforts of many ...

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The integral of $\sec x$ stumped mathematicians in the mid-seventeenth century for quite a while until, in a flash of insight, Isaac Barrow showed that the following can be done: $$\int \sec x \,\mathrm{d}x= \int \frac{1}{\cos x} \, \mathrm{d}x=\int \frac{ \cos x}{\cos^2 x} \, \mathrm{d}x=\int \frac{ \cos x}{1-\sin^2 x} \, \mathrm{d}x.$$ Using $u$-...

64

Theorem: transcendental numbers exist and there are (uncountably) infinitely many of them. The existence of transcendental numbers had been conjectured for over 100 years before Liouville constructed one in 1844. Other numbers such as $e$ were shown to be transcendental one by one. Cantor was able to prove their existence with ease: Proof: the algebraic ...

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Not sure if this satisfies the requirement that we "have no idea what they are", but the extremely strange Mill's constant seems worth mentioning here: There is supposed to be some real number $r>0$ with the property that the integer part of $$r^{3^n}$$ is prime for every natural $n$. It is not known if $r$ is rational and as far as I know not even a ...

33

First of all, the Hodge conjecture is not about one particular differential form. It says that any differential form which satisfies certain conditions will be a $\mathbb{Q}$-linear combination of algebraic forms. There are some particular forms which satisfy those conditions but have not been proven to be such a linear combination.1 However, it is my very-...

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The $\mathcal{AKS}$ (Agrawal, Kayal, Saxena) algorithm, which proves that we can answer if a number is prime or not in polynomial time. It has been found in 2003 and is said "reachable by ordinary man" in reason of the background it needs to be understood. More info here (wiki) and here (the paper).

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In 4 dimensions, it is an open question as to whether there are any exotic smooth structures on the 4-sphere.

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A more or less elementary example I'm quite fond of is the Erdős conjecture on arithmetic progressions, which asserts the following: If for some set $S\subseteq \mathbb{N}$ the sum $$\sum_{s\in S}\frac{1}s$$ diverges, then $S$ contains arbitrarily long arithmetic progressions. I've never seen a heuristic argument one way or the other - I believe ...

26

There are a number of games, like Hex and Chomp, for which it is easy to prove a first player win by strategy stealing but we do not generally know the winning strategy.

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It is not completely elementary, but Abel's proof of the Abel-Ruffini theorem is quite short, 6 pages, and can with a bit of introduction be understood by someone without a degree in mathematics. The Abel-Ruffini theorem states that there is no general solution in radicals to a degree 5 or higher polynomial equation. The Abel-Ruffini theorem had been open ...

25

This is an incredibly difficult problem. It is one of Landau's 4 problems which were presented at the 1912 international congress of mathematicians, all of which remains unsolved today nearly 100 years later.

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This is an inadequate answer, but... These problems are difficult because (a) the space of possible solutions is vast, and (b) there seems insufficient structure to reduce the space so that searching it becomes feasible. Consider the 11-squares problem. Each square has a location and an orientation, and so can be pinned down by specifying three parameters. ...

24

I believe whether or not the Thompson group $F$ is amenable is such question. The paper/article "WHAT is... Thompson's Group" mentions that at a conference devoted to the group there was a poll in which 12 said it was and 12 said it was not. There are in fact papers claiming (at least at the time) to have proofs for both sides. Here are some posts to get an ...

24

Quadratic reciprocity. Euler has stated the theorem but never managed to prove it, and it took Gauss many years to prove this theorem, and right now we have over 200 different proofs, some of which could be explained in an hour long lecture.

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Some people thought for hundreds of years that the Euclidean parallel postulate could be proven from the other four axioms of Euclidean geometry. Giovanni Saccheri even wrote a book about it – Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). However, with the discovery of hyperbolic geometry in 1826 by Nikolai Lobachevsky, the conjecture was ...

22

Take objects which existence proof uses the axiom of choice, e.g: Each vector space has a basis (the standard existence proof uses Zorn's lemma). How does a concrete basis of $C[a,b]$ look like? What about $\mathbb R$ as a $\mathbb Q$-vector space? Ultrafilter, which are used in the construction of the hyperreals: Does the sequence $(0,1,0,1,0,1,\ldots)$ ...

21

Here is an argument that Tate is harder than Hodge: We know the Hodge conjecture in the codimension one case (this is the Lefschetz $(1,1)$ Theorem). On the other hand, the Tate conjecture remains open even in codimension one except in some very special cases. Also, those special cases have often been proved by reducing to the Hodge conjecture. Here ...

21

Hilbert's 10th problem over over $\mathbb{Q}$/Mazur's conjecture. These are two open problems that point in opposite directions, and I think experts really aren't sure which way to guess. Hilbert's 10th problem over $\mathbb{Q}$ Is there an algorithm which, given a collection of polynomial equations with rational coefficients, do they have a rational ...

19

Bhargava and Shankar have proved results about average $3$-Selmer ranks of elliptic curves. (See this arxiv preprint, which is the same paper cited by the Wikipedia article linked in the OP.) Their argument is via geometry of numbers (so to speak). In fact, they are able to construct families such that exactly half of them have positive sign in their ...

18

I improved on Laczkovich's solution by using a different orientation of the 4 small central triangles, by choosing better parameters (x, y) and using fewer triangles for a total of 64 triangles. The original Laczkovich solution uses about 7 trillion triangles. Here's one with 50 triangles:

18

"My current understanding is that the field of one element is the most popular approach to RH." Analytic number theory, with ideas from algebraic geometry, random matrix theory, and any other areas that might be relevant, is the only approach known to have produced any concrete results toward RH. The random matrix theory in particular has produced a lot ...

17

Try Mazur's Questions about Number (1995). One simple, surely fundamental, question has been recently asked (by Masser and Oesterle) as the distillation of some recent history of the subject, and of a good many ancient problems. This question is still unanswered, and goes under the name of the ABC-Conjecture. It has to do with the seemingly ...

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This problem is hard in the sense that it is still unproven. I will provide a set of references, but little conclusive work (as far as I know) has been done on any of them. This is a conjecture of Hardy; he later generalized it to say: if a, b, c are relatively prime, a is positive, and $(a+b)$ and c are not both even, and $b^2 - 4ac$ is not a perfect ...

15

Is $\pi \cdot e$ rational? What is the minimal number of people in a party, such that there are necessarily either at least 5 mutual strangers or at least 5 mutual acquaintances? Is there a positive non-integer $x$ such that both $2^x$ and $3^x$ are integers? Does every closed curve in the plain contain 4 vertices of a square? Can you factor an integer in ...

15

The case of factor $3$ is more interesting because with factor $1$ it's easy to prove that the sequence hits $1$. We can prove this by induction: For $n_0=1,2$ this is clearly true. Suppose that it is true for all starting values $1,2,\dots,n_0-1$. Then if $n_0$ is even, the next number will be $n_0 / 2 < n_0$ and thus the sequence will hit $1$. If $n_0$ ...

15

Here's a wonderful open problem in set theory, which can be translated to a statement which you might be looking for. Suppose that $\aleph_\omega$ is a strong limit cardinal. Is $2^{\aleph_\omega}<\aleph_{\omega_1}$? We can prove that under the assumption that $\aleph_\omega$ is a strong limit cardinal, it is necessarily the case that $2^{\aleph_\... 15 One of my favourites because it's just so simple to state. Discussed in this MO question and related to an also very interesting, though solved question/puzzle which I posted here a while ago. I'm not sure when the problem was first stated but it could certainly have been understood by even the earliest mathematicians. Can a disk be tiled by a finite ... 14 Note that we can write your iteration scheme as, for$i \geq 0\begin{align} x(0,i) & = \text{given sequence of nonnegative integers}\\ x(n+1,i) & = \begin{cases} 0 & i = 0 \\ \left|x(n, i) - x(n,i-1)\right| & i > 0\end{cases}\end{align} The first implication of this definition is that Property 1 the value ofx(n,i)$is ... 14 In the theory of dynamical systems, problems involve limit cycles in general are always very difficult. The second part of Hilbert's sixteenth problem is my personal "favorite". The upper bound for the number of limit cycles of planar polynomial vector fields of degree$n$remains unsolved for any$n>1$. For example, can quadratic plane vector fields ($n=...

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It's not particularly famous, but it should be; something almost all mathematics students encounter without realizing it, the definite integration problem. The problem of whether a given indefinite integral has a closed form antiderivitive expressible in elementary functions is solved, in the form of a semi-algorithm, the Risch algorithm. There is no similar ...

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