Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether, for 1 < k < n - 1, every binomial coefficient n choose k has a prime divisor at most n/2, except 7 choose 3; proved by Ecklund in 1969, while the site's strict bound p < n/2 fails at 4 choose 2.
Asks whether some constant c makes every binomial coefficient in row n have a divisor between c times n and n.
Bounds the largest possible smallest factor when n factorial is written as a product of n increasing factors, in particular whether it approaches n over e.
Asks whether random signs on n unit complex numbers give a sum of absolute value at most the square root of two with probability at least about one over n; Erdős asked it with radius one, which fails for every even n.
Asks whether only finitely many equalities hold between two products of central binomial coefficients taken over distinct indices.
Asks whether a factorial can equal a sum or difference of two kth powers with k greater than two and the powers not both trivial.
Asks whether, for infinitely many n, two numbers whose factorials' product divides n! times the n-th power of the product of the first r primes can sum to more than n plus f(r) log n, with f(r) tending to infinity.
Asks for a proof that every finite set of integers has two members whose greatest common divisor is at most one member divided by the set's size.
Asks whether a power of two can equal a sum of distinct factorials in only finitely many ways.
Asks whether, for each odd prime p, the equation with p minus one factorial plus a power of a equal to a power of p has only finitely many solutions.
Asks whether the number of ways to write an integer as a power of two plus a power of three plus a product of a power of two and a power of three is bounded.
Asks whether infinitely many positive integers are not of the form n minus Euler's totient of n.
The set of limit points of the ratio of the number of divisors of n plus one factorial to the number of divisors of n factorial.
Asks whether there is an increasing sequence of density one all of whose products of consecutive blocks of terms are distinct; answered yes in July 2026 by Chojecki and Sneiderman, accepted by the site, not refereed.
Asks whether the integers eventually produced from two and three by repeatedly adjoining products of two distinct terms minus one have positive density, read as positive lower density following the site's curator.
Asks whether some graph on n vertices has as many as two to the number of vertex pairs divided by n factorial distinct subgraphs occurring in exactly one way.
Asks whether, for every n and d, some run of consecutive primes starting after the n-th prime has sum divisible by d.
Asks whether a sufficiently sparse set missing a residue class modulo every prime must have some shift all of whose members are prime; disproved by Weisenberg's 2024 arbitrarily sparse admissible sets with no prime shift.
Asks whether two infinite sets have a sumset that agrees with the set of primes apart from finitely many exceptions.
Asks whether the largest non-representable integer for the worst coprime k-element subset of the first n integers is asymptotically n squared over k minus one.
Determines which coprime k-element subset of the first n integers leaves the most integers unrepresentable as sums of its members, and whether it is the top k.
The largest integer not expressible as a nonnegative integer combination of the binomial coefficients C(n,1), ..., C(n,n-1), for n not a prime power.
Estimates how many of the partial products of an increasing sequence bounded by x can be perfect squares, and whether nearly x of them can be.
The largest subset of the first N integers whose pairwise sums include no perfect square.
Asks whether every finite coloring of the integers has two distinct integers of one color summing to a square, or to a k-th power; proved by Khalfalah and Szemerédi for every non-constant polynomial with an even value.