Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the smallest set containing one and closed under tripling plus one, doubling plus one, and sextupling plus one has positive lower density.
Asks whether some set of naturals of lower density above one third has no two members, possibly equal, summing to a power of two.
Asks whether an interval of length a constant times the maximal prime gap below x contains the expected number of primes, for y between x halved and x.
Asks whether infinitely many n make n - 2x^2 prime for every x with 2x^2 < n.
Asks whether infinitely many n have n minus k squared prime for every k coprime to n with k squared below n.
Asks whether the set of n with alpha times n squared within one over the logarithm of n of an integer is an additive basis of order two, for irrational alpha.
Asks whether every large integer is a sum of two squares minus a square with all three squares at most that integer.
Determines the density of integers n whose greatest common divisor with the integer part of n to the power alpha is one, for non-integer positive alpha.
Asks whether every plus or minus one polynomial of degree n has maximum modulus on the unit circle above (1+c) times the square root of n for some fixed c>0; answered no by the OpenAI release's construction of 2026-09-23.
Asks a question about the size of the fundamental Lagrange interpolation polynomials determined by n nodes in the interval from minus one to one.
Determines for which orders k the number of permutations of n letters having order exactly k is largest.
Asks whether the logarithm of the radius of the largest origin-centered disc a planar simple random walk covers by time n has order sqrt(log n) in probability; the site's almost-every wording is corrected, and Révész and Dembo–Peres–Rosen–Zeitouni prove it.
Asks the infinitely-often probability of exactly r sites tied for maximum local time in planar simple random walk, for each integer r at least three.
Estimates the least size of a random subset of an abelian group of order N whose subset sums hit every group element nearly equally often.
Asks whether, for each positive epsilon, boundedly many inverses of integers up to p^epsilon represent every residue modulo any prime p; proved by Shparlinski, Croot and Glibichuk, whose bound has order epsilon^(-2).
Asks whether a dense set of integers has a k-term arithmetic progression whose common difference is a difference of two members of any large enough set.
Asks for the size of the supremum of reciprocal sums over finite disjoint congruence families with distinct moduli greater than m; the supremum is known to logarithmic scale.
Asks whether, for a set A of natural numbers and a positive non-decreasing g, the set of n at which the number of representations of n as a sum of two elements of A equals g(n) always has lower density 0, and upper density below some c < 1.
Concerns sets of real numbers of infinite measure in which no ratio of two distinct elements is an integer.
Asks whether the sum of one over a times log a over a primitive set of integers all at least x is at most one plus a quantity tending to zero.
Asks whether, for a set of positive measure and almost every positive x, every large integer multiple of x lies in some integer dilate of the set.
Asks whether every two-coloring of the natural numbers admits an infinite set all of whose sums of products of distinct members share one color; no, by a 1995 theorem of Smith, as the site's thread deduced in 2026.
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 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 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.