Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
1,221 problems
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the sum of one over consecutive products of an increasing integer sequence is irrational whenever the sequence grows doubly exponentially.
Concerns unitary perfect numbers, integers equal to the sum of their proper divisors d for which d and n divided by d are coprime.
Asks whether every k with the sum of divisors of n equal to k times n must be of order smaller than log log n.
Studies the least m such that n is the sum of the k smallest divisors of m for some k, and how that function behaves.
Concerns the classification of primes into classes by repeatedly factoring p plus one, starting from primes whose only such factors are two and three.
Asks whether, for every k at least two, there are a prime and k consecutive intervals of integers whose products are each congruent to one modulo that prime.
Asks whether the number of Carmichael numbers up to x is x to the power one minus a quantity tending to zero.
Asks whether only finitely many n lying between two consecutive primes have n factorial plus one divisible only by the next two primes.
Asks whether infinitely many primes p have p minus k factorial composite for every k with k factorial less than p.
Bounds the number f(n) of integers k with k times the sum of divisors of k equal to n, asking whether f(n) is at most n to a power o(1/log log n), perhaps even a power of log n.
Counts pairs with the sum of divisors of a plus that of b equal to that of a plus b and a plus b at most x, and asks whether the count is proportional to x; Li's June 2026 preprint claims it exceeds x times any log power, pending.
The largest subset of one to n in which no element divides two other distinct elements, and whether its density tends to an irrational limit; an exact formula and an irrational limit near 0.67297, by a Lean proof Conjectures.io certified.
Estimates the least n at least two k for which n minus i divides n choose k for all but one i below k.
Asks whether the totient of n exceeds the totient of n minus its totient for almost all n and the reverse holds infinitely often; the first part proved by Luca and Pomerance (2002), the second by Grytczuk, Luca and Wójtowicz (2001).
Asks whether infinitely many primes are one plus a power of two times a prime, or one plus a power of two times a power of three times a prime.
Concerns the graph on n plane points that are pairwise at least distance one apart, with edges joining the pairs exactly distance one apart.
Asks whether every graph of chromatic number aleph one contains an infinitely connected subgraph of chromatic number aleph one.
Asks whether every graph of chromatic number aleph one contains a countable subgraph that is infinitely vertex-connected.
Bounds the lines containing at least k of n plane points by n squared over k cubed for k up to root n; fails as worded at k = 1, and Szemerédi and Trotter proved it for k from 2 to root n.
Estimates the largest guaranteed number of points, among any n points in the plane, with no two at distance one, and whether it is at least n over four.
Asks whether a finite family of pairwise disjoint unit segments in the unit square can be maximal, so that no further unit segment can be added.
Studies the least n for which a given prime divides n factorial plus one, as a function of that prime.
Asks whether the number of composite numbers below x dividing n factorial plus one for some n is at most x to a power tending to zero.
Asks whether the m for which m factorial plus one has a prime factor not congruent to one modulo m, and the primes arising so (Pillai primes, counted among all primes), have densities, and what they are.
Concerns a constant larger than r to the power minus r, for each r at least three, in a property of r-uniform hypergraphs on many vertices.