Mathematics Problem Archive

Erdős: Suppose I have a sequence of positive integers whose reciprocals sum to infinity. Must that sequence contain arbitrarily long arithmetic progre...

Erdős: In the binary expansion of sqrt(2), are there arbitrarily long sequences of 0's? Can you find a single algebraic number with this property?...

Alon, Peres: Given any subset S of the integers modulo a prime p, what is the least K=K(p) for which there always exists an m so that mS has no gap of...

/Solymosi: Finite field Sylvester-Gallai: Suppose S is a tranversal of Z_(p)^(2), i.e., a set of points in the affine plane so that every row and colu...

Erdős-Turán: Suppose that S is a set of positive integers with the property that S+S -- that is, all sums of the form s_(1)+s_(2) for s_(1), s_(2) in ...

Olson: Every sequence of 2n-1 elements from a group of order n (written multiplicatively) has an n element subsequence with product 1 (in the given or...

Wills, Cusick: Suppose k runners having distinct constant speeds start at a common point and run laps on a unit length circular track. Then for any gi...

Erdős: Suppose that S is a set of positive integers with the property that no element is the sum of a nonempty set of other elements. Such a set is ca...

Jaeger: If F is a finite field with at least 4 elements and A is an invertible n by n matrix over F, then there are vectors x, y in F^(n) which haveal...

Graham: Is x^(2)+y^(2)=z^(2) partition regular? That is, is it true that every coloring of the positive integers by a finite number of colors contains...

Erdős-Strauss : Is it possible, for each positive integer n, to find positive integers a, b, and c so that 4/n = 1/a + 1/b + 1/c ? See this....

Singmaster : Show that there is some B so that no integer appears more than B times among the binomial coefficients. See this....

Carmichael : There is no n so that the only integer m with phi(n) = phi(m) is m=n. ("phi" is the Euler phi/totient function). See this....

Ulam : Is there a dense of points in the real plane so that every two points are at a rational distance? See this....

How quickly do the gaps between successive primes grow? Is it slower than n^(ľ) for every ľ > 0? See this....

Erdős: Is there a prime between n^(2)and (n+1)^(2 )for every n > 0?...

Is the least quadratic residue modulo p at most p^(ľ)^( )for any ľ > 0?...

Choose a random subset $Q$ of the primes by independently retaining each prime with probability $1/2$. Is it almost surely true that $$\limsup_{n\to\i...

Given $\varepsilon>0$, does there exist a polynomial $f(x)=\prod_{i=1}^d(x-\alpha_i)\in\mathbb{Z}[x]$ whose logarithmic Mahler measure $$m(f)=\sum_{i:...

Büchi's problem on sufficiently large sequences of square numbers with constant second difference....

Exponent pair conjecture: for all $\varepsilon > 0$, is the pair $(\varepsilon, 1/2 + \varepsilon)$ an exponent pair?...

Let $K$ be a number field and let $\rho$ be an irreducible $p$-adic representation of $\operatorname{Gal}(\overline K/K)$ that is unramified outside f...

For every totally real number field $F$ and prime $p$, do the Iwasawa invariants $\lambda(F_\infty/F)$ and $\mu(F_\infty/F)$ of the cyclotomic $\mathb...

Hermite's problem: is it possible, for any natural number $n$, to assign a sequence of natural numbers to each real number such that the sequence for ...

Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant $X$ is within a constant multiple o...

For an Artin $L$-function attached to a Galois extension of number fields, is its leading Taylor coefficient at $s=0$ the product of the corresponding...

For a motive (or the cohomology of a smooth projective variety) and an appropriate integer argument, is the order of vanishing of its L-function the p...

Find the value of the De Bruijn–Newman constant....

For every $\varepsilon>0$, is there a $T_0(\varepsilon)$ such that, whenever $T\geq T_0$ and $H=T^{1/4+\varepsilon}$, the interval $(T,T+H]$ contains ...

For fixed admissible $k$, does $T^{-1}\int_0^T|\zeta(1/2+it)|^{2k}\,dt$ have the Keating–Snaith asymptotic $a(k)G(k+1)^2G(2k+1)^{-1}(\log T)^{k^2}$, w...

The density hypothesis for zeroes of the Riemann zeta function....

Piltz divisor problem on bounding $\Delta_k(x) = D_k(x) - xP_k(\log(x))$...

Let $K$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$ with unitary central character. Is ev...

Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $1/4$....

Bombieri–Lang conjecture: $K$-rational points on a variety of general type over a number field $K$ are not a dense set in Zariski topology....

Manin conjecture: if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: $H(x)\leq B$ in any Zarisk...

For an abelian variety or suitable motive over a number field, are its normalized Frobenius conjugacy classes equidistributed in the associated compac...

Vojta's conjecture: points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in s...

Fix $n\geq3$. If coprime nonzero integers $a_1,\ldots,a_n$ have sum zero and no proper subsum zero, is it true that for every $\varepsilon>0$ there is...

Szpiro's conjecture: for any $\varepsilon > 0$, there is some constant $C(\varepsilon)$ such that, for any elliptic curve $E$ defined over $\mathbb{Q}...

Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are c...

Erdős–Moser problem: is $1^1 + 2^1 = 3^1$ the only solution to the Erdős–Moser equation?...

Which integers can be written as the sum of three perfect cubes?...

For coprime integers $1\leq a<d$, is the least prime $p(a,d)$ congruent to $a\pmod d$ always less than $d^2$?...

Are there infinitely many palindromic primes to every base?...

For every number field $K$ and every prime $\ell$, does the Iwasawa invariant $\mu_\ell(K)$ vanish?...

For every odd prime $p$ and every positive integer $t$, does there exist a $p$-rational number field $K$ with $\operatorname{Gal}(K/\mathbb Q)\cong(\m...

For a finite abelian extension of number fields, does the Brumer–Stark element have the predicted ideal-class annihilation and principalization proper...

If $k$ is a number field that is not totally real, is there a hyperbolic $3$-manifold with trace field $k$?...