Unsolved Problems

Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $\frac{1}{2}$?...

The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1...

Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For...

Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this ...

Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)....

Every even integer greater than 2 can be expressed as the sum of two primes....

For any $\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \text{rad}(abc...

For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$....

Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?...

Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?...

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1....

Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field....

Are there infinitely many primes of the form $n^2 + 1$?...

Find all integer solutions to $n! + 1 = m^2$....

For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z....

If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...

Generalize the reciprocity law of number theory to arbitrary number fields....

Extend the theory of quadratic forms with algebraic numerical coefficients....

Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely ...

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?...

Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?...

Suppose $A, B \subset \{1, \dots, N\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?...

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 $|A + A| \geq |A|^{1+c}$?...

Suppose $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1-o(1)}$?...

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...

Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?...

Is there $c > 0$ such that whenever $A \subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?...

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....

Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\theta$ such that no interval of length $\frac{1}{n}$ in $\mathbb{R}/\mat...

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...

Let $N$ be large. For each prime $p$ with $N^{0.51} \leq p < 2N^{0.51}$, pick a residue $a(p) \in \mathbb{Z}/p\mathbb{Z}$. Is $\#\{n \in [N] : n \equi...

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leq p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have s...

Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them?...

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...

Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$....

Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?...

Are there infinitely many primes $p$ such that $2p + 1$ is also prime?...

For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?...

For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?...

For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$?...

What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?...

Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)...

Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?...