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 some bounded r makes the integers of the form a power of two plus a number with at most r prime divisors have density at least one minus epsilon.
Estimates the longest run of pairwise distinct consecutive prime gaps starting at an index below x, asking whether it exceeds a power of log x and whether it is o(log x).
Asks whether the smallest even number missing from the first x prime gaps tends to infinity, and whether it grows faster than log x.
Asks about the arithmetic of the primorials, the products of the first k primes.
Asks whether the number of primes up to x plus y is at most the number up to x plus the number up to y, for all large x and y.
Estimates, for k at least three, the largest reciprocal sum of a set of integers up to N with no k members sharing the same pairwise least common multiple.
Estimates the least number of subsets of the integers up to n that forces a sunflower of size k, meaning k of them with equal pairwise intersections.
Estimates the largest reciprocal sum, over the logarithm of N, of a set of integers up to N in which no member equals another member times a factor whose prime factors all exceed the smaller member.
Asks whether the density of the integers n for which a given t is a sum of distinct divisors of n is asymptotic to a constant over a power of log t; false by a Lean disproof the bounty site Conjectures.io certified in 2026.
Estimates the shortest interval length that always contains distinct integers, one divisible by each prime up to n.
Asks how many Sidon subsets of the integers up to N there are, compared with two to the power of the largest Sidon set size.
Asks whether the number of maximal Sidon subsets of the integers up to N is below two to the power o(square root of N), and whether it exceeds two to the power N^c for some c > 0.
Asks whether, for r at least 2, the largest sets up to N with at most r representations of each sum, and of each positive difference, have different square-root constants, and whether the difference constant is the smaller.
Estimates the largest set of integers up to N with at most one number having more than one representation as a sum of two members.
Asks whether every set of integers up to N of size just above five eighths of N contains three members whose three pairwise sums also lie in the set; proved in a 2026 preprint, developed with GPT-5.5 Pro, that the site accepted.
Estimates, for k at least three, how far above N a subset of the integers up to 2N must be to force k integers whose pairwise sums all lie in the set; open, with the thresholds 1 and 3 for k = 3 and 4 and bounded for k = 5.
Asks whether a set of integers up to N in which no sum of consecutive members lies in the set has size at most half of N plus a constant; false, by Freud's 1993 construction of density 19/36.
Asks whether an additive basis of order two whose representation counts tend to infinity must contain a minimal such basis.
Asks whether the union of two disjoint additive bases of order two must contain a minimal additive basis of order two.
Asks whether representation counts growing like a constant times the logarithm force an additive basis of order k to contain a minimal one, for k at least 3.
Asks whether an additive basis of order two whose representation counts tend to infinity can be split into two disjoint additive bases of order two.
How long the game in which two players alternately add integers up to n to a shared set free of divisibility between members can be guaranteed to last, and whether it lasts at least εn or (1-ε)n/2 moves.
Asks whether, for every positive epsilon, some k makes the number of windows of k consecutive members with least common multiple below X less than X to epsilon.
Estimates the largest set of integers up to N whose sets of sums of r distinct members are disjoint for distinct r, and whether it nears two root N.
Determines how slowly an infinite set of naturals can grow while its sets of sums of r distinct members stay disjoint for different r.