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 a set of n integers up to N whose subset sums are all distinct forces N to be at least a constant times 2 to the power n.
Asks whether finite distinct covering systems can have arbitrarily large minimum modulus; Hough proved an absolute bound.
Asks whether every set of natural numbers whose reciprocals sum to infinity must contain arbitrarily long arithmetic progressions.
Asks whether prime gaps exceed any given constant times log n times a slowly growing factor built from repeated logarithms infinitely often.
Asks whether, for every constant C at least 0, the gap after the nth prime divided by log n tends to exactly C along some sequence of indices n.
Asks whether there are infinitely many n for which three consecutive prime gaps are strictly increasing.
Asks whether there is a covering system of congruences whose moduli are distinct and all odd.
Asks whether every finite coloring of the integers admits a covering system whose moduli all receive the same color.
Asks whether the odd integers that are not the sum of a prime and two powers of 2 have positive upper density.
Asks whether some fixed k makes every large integer the sum of a prime and at most k powers of 2; open, with three powers shown insufficient for infinitely many even integers.
Asks whether every large odd integer is the sum of a squarefree number and a power of 2.
Asks how large an infinite set with no element dividing the sum of two larger elements can be, in counting, density and reciprocal-sum terms; the first two questions have 2026 claimed answers, the reciprocal sum is open.
Asks whether a subset of the first N integers with no element dividing the sum of two larger elements has size at most N over 3 plus a constant; proved by Bedert in 2023, with the ceiling of N over 3 exact for large N.
Asks whether the integers up to N lacking a unique representation as a sum of two elements of a set must number nearly the square root of N; yes, and never little-o of it, by two Lean proofs the bounty site Conjectures.io accepted.
Asks whether the alternating sum of n divided by the nth prime converges; open, with Tao's proof of convergence conditional on a strong Hardy-Littlewood prime tuples conjecture.
Asks whether the odd integers not of the form a power of 2 plus a prime form the union of an infinite arithmetic progression and a set of density zero.
Asks whether infinitely many primes p have every even number up to p minus 3 expressible as a difference of two primes not exceeding p.
Asks how few distinct divisors of a practical number represent every smaller integer, in particular for factorials; h(n!) < n^{o(1)} is proved (Conjectures.io, 2026), the (log log m)^{O(1)} part claimed, the rest open.
Asks whether a graph made of n edge-disjoint copies of the complete graph on n vertices has chromatic number exactly n.
Asks whether the number of n-element sets needed to force a k-sunflower grows only exponentially in n, with a base depending on k.
Asks whether the smallest intersecting family of n-element sets in which every set of size at most n minus 1 misses a member has size linear in n.
Asks whether some graph on n vertices has at least n squared over 8 edges, no complete subgraph on 4 vertices, and no large independent set.
Asks whether every triangle-free graph on 5n vertices can be made bipartite by deleting at most n squared edges.
Asks whether every triangle-free graph on 5n vertices contains at most n to the fifth power many 5-cycles.
Asks whether the integers avoiding a chosen residue class for each modulus in an increasing sequence always have a logarithmic density.