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...

3n+1 ("Collatz" or "Ulam") problem: Take any positive integer, and apply the following process: (1) divide it by two if it's even, multiply by three a...

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?...

Are the positive integer powers of 3/2 mod 1 uniformly distributed in the unit interval? One would think so, but apparently this is a hard question. S...

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...

Niederreiter: Show that there exists a B so that, for every n > 0, there exists a k relatively prime to n whose continued fraction has partial quotien...

/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...

Dudeney: Is it possible to choose 2n points in an n by n grid in the plane so that no three are collinear? Conjecture: no. In fact, it is conjectured ...

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...

Rado: Is it true that, for every n, there is an integer M(n), so that whenever a linear homogeneous equation in n variables is Ramsey (in the positive...

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?...

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...

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:...

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?...

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 ...

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...

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

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...