166

One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?
update 2012-07-12 Since the question has returned to the front page, I'm taking the liberty to add some links that ...

129

The moving sofa problem: What rigid two-dimensional shape has the largest area $A$ that can be maneuvered through an L-shaped planar region with legs of unit width?
So far the best results are $2.219531669\lt A\lt 2.37$.

109

This is the second time I've seen this question on MathOverflow and this will be the second time I've posted this answer.
Singmaster's conjecture says there is a finite upper bound on the number of times a number (other than the $1$s on the edge) can appear in Pascal's triangle. The upper bound may be as low as $8$. If so, then no number (besides those $1$...

104

The lonely runner conjecture. As Wikipedia puts it:
Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be lonely if at distance of at least $1/(k + 1)$ from each other runner. The lonely runner conjecture states that ...

94

The Casas-Alvero conjecture: let the characteristic of the field $k$ be $0$. If a monic polynomial $f\in k[X]$ of degree $n$ has a common root with each of its derivatives $f',\ldots,f^{(n-1)}$, then $f(X)=(X-a)^n$ for some $a\in k$.

91

Gourevitch's conjecture:
$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$

85

There is a lot of number theory elementary conjectures, but one that is especially elementary is the so called Giuga Conjecture (or Agoh-Giuga Conjecture), from the 1950:
a positive integer $p>1$ is prime if and only if
$$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod{p}$$

84

Is the sequence $(3/2)^n \mod 1$ dense in the unit interval?
In the other direction, Mahler's 3/2 problem:
Do all elements of this sequence with large enough index $n$ lie in the interval $(0,1/2)$?
It is known that $\beta^n$ is uniformly distributed modulo one for almost all $\beta>1$, but explicit examples of $\beta$ for which density holds are ...

75

I find George Dantzig's story particularly impressive and inspiring.
While he was a graduate student at UC Berkeley, near the beginning of
a class for which Dantzig was late, professor Jerzy Neyman wrote two
examples of famously unsolved statistics problems on the blackboard.
When Dantzig arrived, he assumed that the two problems were a homework
...

74

Is $e+\pi $ rational?

70

From "An Invitation to Mathematics":
Are there any integer solutions to $x^3 + y^3 + z^3 = 33$?
I thought this might be a good candidate since that book was meant as a bridge from competitive Mathematics to research. There are a few other examples, but I am quoting only one here due to your requirement.
Edit: Such integers x, y and z have been found.

69

It is currently unknown if all triangles have a periodic billiard path. (See, for example, http://en.wikipedia.org/wiki/Outer_billiard#Existence_of_Periodic_Orbits)

63

I always enjoyed telling people about the Inscribed square problem :
Does every (Jordan) curve in the plane contain all four vertices of
some square?
Update: Here is a variation due to Helge Tverberg: Does every (polygonal) curve in the plane outside of the unit circle, contain all four vertices of some square with side length >0.1? This version ...

62

Yes, there are several. Here's a few which I personally care about (described in varying amounts of precision). This is not meant to be an exhaustive list, and reflects my own biases and interests.
I am focusing here on questions which have been open for a long amount of time, rather than questions which have only recently been raised, in the hopes that ...

61

Jonas Meyer's comment:
Quote from arxiv.org/abs/0902.3961, Bjorn Poonen, Feb. 2009: "Harvey Friedman asked whether there exists a polynomial $f(x,y)\in Q[x,y]$ such that the induced map $Q × Q\to Q$ is injective. Heuristics suggest that most sufficiently complicated polynomials should do the trick. Don Zagier has speculated that a polynomial as simple as $x^...

57

There are infinitely many primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$.
First explicitly asked by Gauss, now generally thought of as a corollary of Artin's primitive root conjecture.

55

Problem: The partition function $p(n)$ is even (resp. odd) half of the time.
Of course you need to explain to a general audience what the partition function is, but that's not hard, my daughter in K1 got as an assignment to compute $p(n)$ for $n$ up to 4.
You also need to explain "half of the time", which means that the number of $n < x$ such that $p(n)$ ...

55

A proof of this conjecture of Erdos would certainly turn heads, raise eyebrows, and garner the attention of the Fields Medal committee.
If $\sum_{a \in A} \frac 1a$ diverges and $A\subseteq {\mathbb N}_{>0}$, then $A$ contains a 3-term arithmetic progression.
Probably "diverges" can be replaced with "is bigger than 4".

54

The circulant Hadamard matrix conjecture, first stated in print by Ryser in 1963. It can be stated as follows. If $n>4$, then there does not exist a sequence $(a_1,a_2,\dots,a_n)$ of $\pm 1$'s satisfying
$$ \sum_{i=1}^n a_i a_{i+k}=0,\ 1\leq k\leq n-1, $$
where the subscript $i+k$ is taken modulo $n$.

53

The question is still open. There are at least two versions. The most popular asks for the minimal-area convex subset of the plane such that every set with diameter 1 can be translated, rotated and/or reflected to fit inside it. Here is the best lower bound I know:
Peter Brass and Mehrbod Sharifi, A lower bound for Lebesgue's universal cover problem, Int....

53

8 Lonely Runners
[The aim of this proposal would be to find a project that a massive number of people (including amateur mathematicians) might actually effectively contribute to, which is a somewhat different goal than the other proposed polymath projects.]
A longstanding problem in diophantine approximation is the lonely runner conjecture which states:
...

52

Michael Atiyah posted a short paper "The Non-Existent Complex 6-Sphere" https://arxiv.org/abs/1610.09366 with a claimed negative solution to the problem.

50

At the risk of stretching my own rule, please allow that I could define "ring" for a high school senior. Then I'd proffer this question I heard years ago from Melvin Henriksen:
Must a non-commutative ring (with identity) contain a non-zero-divisor aside from the identity?

49

Is every algebraic curve in $\mathbb P^3$ the set-theoretic intersection of two algebraic surfaces ? Not known!

47

Sendov's Conjecture
For a polynomial $$f(z) = (z-r_{1}) \cdot (z-r_{2}) \cdots (z-r_{n}) \quad \text{for} \ \ \ \ n \geq 2$$ with all roots $r_{1}, ..., r_{n}$ inside the closed unit disk $|z| \leq 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point of $f$.

46

Here is one which I found at this MO link:
$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan(t)+\sqrt{7}}{\tan(t)-\sqrt{7}}\right|\ dt = \sum_{n\geq
1} \left(\frac n7\right)\frac{1}{n^2}, $$
where $\displaystyle\left(\frac n7\right)$ denotes the Legendre symbol. Not really
my favorite identity, but it ...

45

The irrationality of Catalan's constant $G=1-1/3^2+1/5^2-1/7^2+\cdots$.
Remarks: Although Catalan's constant is certainly well-known, the irrationality is
the tip of the iceberg of a related conjecture of Milnor about the linear independence
over the rationals of volumes of certain hyperbolic 3-manifolds (which is a special
case of a conjecture of ...

45

Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: https://arxiv.org/abs/1812.10454.
This was probably considered the biggest open problem in the combinatorics of simplicial complexes. See Gil Kalai's blog post: https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-...

42

The circulant Hadamard matrix conjecture states that for $n>4$ there
does not exist a sequence $(a_1,\dots,a_n)$ of $\pm 1$'s that is
orthogonal to every proper cyclic shift of itself. It has a similar
flavor to the Erdős discrepancy problem that was the topic of
Polymath5. Terry Tao says the following on his blog about the
circulant Hadamard matrix ...

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