Mathematics Problem Archive

Explicitly construct test elements or principal homogeneous spaces having prescribed ramification and a prescribed large prime order $\ell$....

Work efficiently with the testing characters and principal homogeneous spaces without constructing them explicitly....

Identify and solve tractable special cases of the signature problem described in the slides....

For nearby moduli $n_i=n_1+d_i$ and maps $f_i(r)=r^{e_i}\bmod n_i$, prove the conjecture that with sufficiently many components at least one $f_i$ is ...

For a prime $p$, conjecturally $p$ does not divide the class number of the maximal real subfield $\mathbb{Q}(\zeta_p+\overline{\zeta_p})$ of the $p$th...

Let $\zeta_p:\mathbb{Z}_p\to\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\zeta_p(k)\ne0$ for every even integer $k$?...

Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ i...

Is every integer $n\equiv5,6,$ or $7\pmod 8$ a congruent number?...

For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?...

Given an elliptic curve $E:y^2=x^3+Ax+B$ over $\mathbb{Q}$, determine whether $E$ has infinitely many rational points....

For an elliptic curve $E/\mathbb{Q}$, let $N_p$ be its number of solutions modulo $p$ plus one and put $f(X)=\sum_{p\le X}\log(N_p/p)$. Is $f(X)$ boun...

For an elliptic curve $E/\mathbb{Q}$ of rank $r$, does $f(X)=\sum_{p\le X}\log(N_p/p)$ grow asymptotically like $r\log\log X$?...

Find a higher-dimensional analogue of the paper's cropping and summation argument; in dimension three the expected sum ranges over quadruples $v_1,v_2...

For $Z(s)=\sum f(a,b,c,d)^s$, which is known to converge for real $s>1/2$, extend $Z$ to complex values of $s$....

Can the function $f$ on $SL(2,\mathbb{Z})$ be extended naturally to $\mathbb{C}/SL(2,\mathbb{Z})$? Can analogous series be constructed for other latti...

For every odd prime $p$ and $k\ge1$, is $s(p^k)=k$? For $k\ge2$, is $d(p^k)=p^{k-1}d(p)$?...

For every $k\ge1$, is $s(2^k)=u_k$? Is $d(2^k)=2^k$ for $k\ne2$, with $d(4)=2$?...

Is Arnold's sequence $(u_k)_{k\ge1}$ the $f$-transform of the quadruple $(2,4,4,4)$?...

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+...

Given a prime modulus $p$, a group generator $g$, and the public values $g^a$ and $g^b$ modulo $p$, can the shared value $g^{ab}\bmod p$ be computed e...

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

Given a unit square, does there exist a point in its plane, inside or outside the square, whose distances from all four vertices are rational? Equival...

Find the two prime factors of the RSA-1024 challenge integer $1350664108659952233496032162788059699388814756056670275244851438515265106048595338339402...

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 ...

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....

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?...

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

Characterize all algebraic number fields that have some power basis....

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

Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class lie on the critical line $1/2 + it$ with r...

Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class....

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...

If the base-$b$ expansion of a real number can be emitted in real time by a multitape Turing machine (bounded time between successive digits), must th...

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 $2$....

Agrawal's conjecture that given coprime positive integers $n$ and $r$, if $(X - 1)^n \equiv X^n - 1 \pmod{n, X^r - 1}$, then either $n$ is prime or $n...

Dubner's conjecture: every even number greater than $4208$ is the sum of two primes which both have a twin....

Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful....