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...

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...

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:...

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

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

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

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

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