Mathematics Problem Archive

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....

Erdős: What is Σ_(n≥1 )φ(n)/2^(n), where φ(n) is the Euler phi (totient) function, counting the number of integers less than n which are relatively pr...

/Riasanovsky: Let f be the formal power series over Z/2Z whose nth coefficient is the parity of the divisor function Ă_(0)(n). Is it true that the den...

: Let f be the formal power series over Z/2Z whose nth coefficient is independently chosen to be 1 with probability Ü(n^(-2)) (and probability 1 for n...

Do there exist positive integers $a,b,c,d$ such that all six pairwise face diagonals $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{a^2+d^2}$, $\sqrt{b^2+...

Can Fermat's Last Theorem be proved using only mathematical techniques that were available in the seventeenth century?...

Does there exist a $3\times3$ semi-magic square whose nine entries are distinct positive integer cubes and whose three row sums and three column sums ...

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

Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than $1$?...

Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating....

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

Are there any pairs of betrothed numbers which have same parity?...

Are there any pairs of relatively prime amicable numbers?...

Are there infinitely many betrothed numbers?...

Do any odd noncototients exist?...

Do any (2, 5)-perfect numbers exist?...

Do any Taxicab(5, 2, n) exist for n > 1?...

Is every positive integer expressible as a sum of at most five tetrahedral numbers?...

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 ...

Kummer–Vandiver conjecture: primes $p$ do not divide the class number of the maximal real subfield of the $p$-th cyclotomic field....

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....

Is Selberg class of Dirichlet series equal to class of automorphic L-functions?...

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}...

Zilber–Pink conjecture that if $X$ is a mixed Shimura variety or semiabelian variety defined over $\mathbb{C}$, and $V \subseteq X$ is a subvariety, t...

Can a discrete logarithm on a elliptic curve be computed in sub-exponential time?...

Does every rational number with an odd denominator have an odd greedy expansion?...

Which transcendental numbers are (exponential) periods?...

How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such a...

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

Goormaghtigh conjecture on solutions to $(x^m - 1)/(x - 1) = (y^n - 1)/(y - 1)$ where $x > y > 1$ and $m, n > 2$....

The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exactly one normalized solution to the Markov Diophanti...

Pillai's conjecture: for any $A, B, C$, the equation $Ax^m - By^n = C$ has finitely many solutions when $m, n$ are not both $2$....