Unsolved Problems

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?...

How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?...

Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?...

For any $\varepsilon > 0$, is there a constant $c(\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\varepsilon) x^{1/2-\varepsilon}$?...

If Euler's totient function $\phi(n)$ divides $n-1$, must $n$ be prime?...

Does there exist a 3×3 magic square composed entirely of distinct perfect squares?...

Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?...

Does the partition function satisfy any arbitrary congruence infinitely often?...

Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?...

Are there infinitely many perfect numbers?...

Do quasiperfect numbers exist?...

Do any almost perfect numbers exist that are not powers of 2?...

Are there exactly 65 idoneal numbers, or could there be 66 or 67?...

Do any pairs of amicable numbers exist where one is odd and one is even?...

Are there infinitely many pairs of amicable numbers?...

Are there infinitely many Giuga numbers?...

Do Lychrel numbers exist in base 10?...

Do any odd weird numbers exist?...

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

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

Is 10 a solitary number (no other number shares its abundancy index)?...

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

If $B$ is an additive basis of order 2, must the representation function tend to infinity?...

If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \geq k$?...

Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?...

Does every nonnegative integer appear in Recamán's sequence?...

Can an algorithm determine if a constant-recursive sequence contains a zero?...

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

Do the Ulam numbers have a positive density?...

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 $\liminf_{n \to \infty} n \|n\alpha\| \|n\beta\| = 0$?...

Can integer factorization be solved in polynomial time on a classical computer?...

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?...

Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?...