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.
Bounds by n to the power three halves the edges of an n-vertex graph avoiding a fixed graph made of a vertex joined to k others whose pairs are linked.
Asks whether the largest number of distinct clique sizes in a graph on n vertices is n minus log_2 n minus the iterated-logarithm count, up to O(1); disproved by Spencer, whose construction removes the iterated term.
Asks whether the density exists of integers n whose largest prime factor is below n to the alpha while that of n plus 1 is below n plus 1 to the beta; the OpenAI release of September 2026 shows it is a Dickman product, accepted on its built Lean proof.
Asks for the least prime cutoff x such that a positive density of blocks of k consecutive integers have every member divisible by a prime up to x; the inverse of Problem 687's covering function, open above the square root of k.
Asks whether, for every r, some k makes the product of all integers in any r disjoint intervals of length at least k never a perfect power.
Asks whether only finitely many disjoint blocks of consecutive integers, of lengths k1 and k2 at least 3, have products with the same prime factors.
Asks whether infinitely many prime gaps contain at least two integers all of whose prime factors are smaller than the length of that gap.
Asks whether the largest divisor of n times n plus 1 built only from the primes 2 and 3 exceeds any fixed multiple of n log n for suitable n.
The least number of edges forcing a graph of maximum degree at most d to have two edges at distance at least t; open, exact for t = 1, t = 2 and h_3(3) = 23, between 0.629^t d^t and 3d^t/2 + 1 in general, with 2026 preprints at t = 3.
Asks whether the powerful part of n(n+1)...(n+l) is below n^(2+eps) for every eps once n is large, whether its ratio to n squared is unbounded when l is at least 2, and whether its ratio to n^(l+1) tends to zero.
Asks whether 2 to the n plus or minus 1 and n factorial plus or minus 1 are powerful numbers for only finitely many n.
Asks whether there are infinitely many four-term arithmetic progressions made of pairwise coprime powerful numbers.
Concerns the increasing sequence of powerful numbers, those integers divisible by the square of every prime dividing them, and the gaps between them.
Concerns sums of coprime r-powerful numbers; the 3-powerful triple question is answered yes (Nitaj 1995), infinitely many solutions exist for every r at least 6, and r = 4 is open.
Concerns r-powerful numbers for r at least 3, the integers divisible by the r-th power of each of their prime factors.
Asks whether every sufficiently large integer is the sum of at most three powerful numbers.
Estimates how many powerful integers lie between consecutive squares, in particular whether some fixed power of log n bounds that count for every n and is nearly reached for infinitely many n.
Asks whether the number of ways of writing n as a sum of two powerful numbers is smaller than any fixed power of n.
Asks whether for every k at least 4 and r at least 1 some k-chromatic graph has every vertex critical and no critical set of at most r edges; settled by a Lean proof the bounty site Conjectures.io certified in September 2026.
Estimates the longest run of consecutive integers below x with all divisor counts distinct, and whether short intervals must repeat a divisor count.
Asks whether there are infinitely many n for which n and n plus 1 have the same number of divisors.
Asks whether an exact covering system exists, finitely many congruence classes with distinct moduli such that every integer satisfies exactly one of them.
Asks whether some bound and some number of colors force, in every coloring of the integers, a slowly growing sequence whose subset sums miss a color; no, by a 2026 AI-generated coloring accepted by the site after review.
Asks whether the complement of any sum-free set of reals contains a set of size continuum whose pairwise sums also avoid it; open; AlphaProof's Lean proof of the Sidon case is a pending claim and a thread comment argues the Baire case.
Asks for the limit inferior, limit superior and growth of the sum of the reciprocals of n minus p taken over all primes p below n.