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, for every epsilon and eta, some k makes the largest prime factor of n(n+1)...(n+k) exceed n to the one minus epsilon for a set of n of density at least 1 - eta, the density read as lower density following the site's curator.
Asks whether some k exists so that sieving half the residue classes modulo each of k primes below n to the one minus epsilon leaves few integers up to n.
Asks whether the maximum over k of the number of distinct prime factors of n plus k times log log k over log k tends to infinity as n grows.
Estimates the smallest largest element, and smallest average, of k integers missing a residue class modulo every prime; the largest element lies between half and once k log k, the average between a quarter and half of it; open.
The largest number of congruences guaranteed, over choices of one residue class modulo each n up to x, to be satisfied by every integer up to x; the site's own argument gives log x plus lower-order terms, a pending claim.
Asks whether the set of cubes up to N cubed contains a Sidon set, with all pairwise sums distinct, of size proportional to N.
Estimates the largest subset with no isosceles triangle guaranteed in any n points in d dimensions, in particular whether in the plane it is below a power of n.
Estimates, for fixed dimension d, the largest number of points with all distances distinct that must lie inside every set of n points in d-dimensional space.
Asks whether one shift n making every n plus a term of a fast-growing sequence prime forces infinitely many, with squarefree and doubly exponential variants; three questions answered no, three stay open.
Asks whether a set of pairwise coprime integers below n has the sum of one over n minus a at most the sum of reciprocals of primes below n plus a constant.
Asks how large the larger upper logarithmic density of the two subset-sum sets must be when the natural numbers are split into two classes; Conlon, Fox and Pham determined the minimum as (2 + sqrt 3)/4.
Asks for a path to infinity through coprime pairs above 1 with a composite coordinate, steps changing one coordinate by one; Erdős first asked it without the composite condition, which Stewart quickly answered yes.
Asks whether, for every starting value a and gap bound K, any long enough integer sequence starting at a with gaps at most K has two intervals with equal sums; yes by Hegyvári's 1986 Theorem 3, with an explicit bound.
Asks whether two positive integers x and y must be equal when the primes dividing x to the n minus one match those dividing y to the n minus one for every n.
Asks whether a constant bounds the length of a path from zero to the unit circle inside the region where such a polynomial has modulus below one.
Estimates the largest transitive subtournament every tournament on n vertices must contain; floor(log_2 n) + 1 first fails at n = 14 and fails for infinitely many n; f is known exactly for n <= 33 and 47 <= n <= 56.
Asks whether a sequence of positive lower logarithmic density contains a divisibility chain whose upper growth rate against log log x is at least the weighted sum's; answered yes in 2026 by Alexeev and seven coauthors.
Asks whether, under GCH, the successor of aleph_{omega_{omega+1}} fails the partition relation for triples whose first target is that cardinal and whose other countably many targets are 4.
Asks whether a sum of strictly increasing powers two to the aleph n_k, the first above aleph omega, satisfies the partition relation to aleph omega for pairs.
Asks whether a singular cardinal that is aleph-zero-inaccessible, as is its cofinality, satisfies the partition relation to itself and aleph one for pairs.
Asks whether, for a sequence on the circle, the de Bruijn–Erdős constants for the largest and smallest sums of r consecutive gaps and for their ratio deviate from their trivial values by more than any constant over r as r grows; a 2026 preprint claims all three parts for distinct points, unrefereed and unreviewed.