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...

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...

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

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

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 $n$?...

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.)...

If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?...

For how many prime numbers $p$ is a given integer $a$ (not $\pm 1$ or a perfect square) a primitive root modulo $p$?...

For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?...

For which number fields is there an algorithm to determine solvability of Diophantine equations?...

Is $\pi$ a normal number in base 10?...

Are all irrational algebraic numbers normal in every base?...

If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?...

What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?...

Are there infinitely many real quadratic number fields with unique factorization?...

Can the Kronecker-Weber theorem on abelian extensions of $\mathbb{Q}$ be extended to any base number field?...

Does the $p$-adic regulator of an algebraic number field not vanish?...

For all $\varepsilon > 0$, does $\zeta(1/2 + it) = o(t^\varepsilon)$ as $t \to \infty$?...

Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?...

Do all automorphic L-functions have their nontrivial zeros on the critical line?...

Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?...

What is the optimal exponent in the error term for the divisor summatory function?...

If $x_1, x_2$ are linearly independent over $\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e...

Is the Euler-Mascheroni constant $\gamma$ irrational?...

Is $\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \cdots$ transcendental?...

For any two real numbers $\alpha, \beta$, does $\liminf_{n \to \infty} n \|n\alpha\| \|n\beta\| = 0$?...

For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?...

Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?...

Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?...

Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?...

Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?...

Do any odd perfect numbers exist?...

Are there infinitely many perfect numbers?...

Is π a normal number (all digits equally frequent in all bases)?...

Are all irrational algebraic numbers normal?...

Are there infinitely many real quadratic fields with class number 1 (unique factorization)?...

Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields....

Does the p-adic regulator of an algebraic number field never vanish?...

Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?...

For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?...

Is the Euler-Mascheroni constant γ irrational? Transcendental?...

For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?...

If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?...

Can integer factorization be done in polynomial time?...