Unsolved Problems

Is every $n \leq N$ the sum of two integers, all of whose prime factors are at most $N^\varepsilon$?...

Is there an absolute constant $c > 0$ such that if $A \subset \mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \geq |A|^{1+c}$?...

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \dots, p-1\}$ congruent to some product $a_1a_2$ mo...

Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \in A$, then $a_1a_2 - 1 \in A$). Does $A$ h...

Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$?...

Determine bounds for Waring's problem over finite fields....

Is a random polynomial with coefficients in $\{0, 1\}$ and nonzero constant term almost surely irreducible?...

Let $d \geq 3$ be odd. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ then any homogeneous polynomial $F(\mathbf{x}) \in \mathbb{Z}[x_1, \dots, x_n...

Finding a single solution to $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roug...

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$?...

Let $d \geq 3$ be an odd integer. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ the following is true: given any homogeneous polynomial $F(\mathbf...

Finding a single solution to a polynomial equation $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of su...

What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\prod_{i=1}^{k_1} (a+i) \equiv \prod_{i=1}^{k_2} (a+k_1+i) \equiv \p...

What is the least positive odd number not of the form $\pm p^a \pm 2^b$, where $p$ is an odd prime?...

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $ \lvert x_{n+1}-x_n\rvert \ll 1? $ ...

Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for...

Problem Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $$ m(A) = - \min_x \sum_{a \in A} \cos(ax). $$ What is $m(n) = \min_A...

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational. Conjecture There doe...

Conjecture $\pi$ and $e$ are algebraically independent...

Question Is Euler-Mascheroni constant an transcendental number?...

Conjecture Are all Fermat Numbers $$ F_n = 2^{2^{n } } + 1 $$ Square-Free?...

Conjecture A Fermat prime is a Fermat number $$ F_n = 2^{2^n } + 1 $$ that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_...

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?...

Say that a set $S \subseteq {\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums. Conjecture There exists a fixed cons...

Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1....

For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elemen...

Conjecture Let $G$ be an abelian group of odd order and let $A,B \subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered...