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 every sufficiently large integer has the form a times a prime squared plus b, with a at least one and b less than that prime.
Asks whether two disjoint blocks of k consecutive integers can have the same least common multiple; Erdős's 1979 conjecture that they cannot is open, with a few solutions known only when the block lengths differ.
Asks whether infinitely often a block of k consecutive integers has a larger least common multiple than a later block of k+1; proved in a strong form by Cambie (2024, arXiv), the ratio exceeding any constant.
Asks whether infinitely many n have every n minus k with fewer distinct prime factors than about the logarithm of k over its own logarithm.
Asks whether every sufficiently large n admits some k for which the least prime factor of n plus k exceeds k squared plus one.
Asks whether every large n has some k for which n plus k is composite and the least prime factor of n plus k exceeds k squared.
Asks whether almost every n admits an integer strictly between the nth and next prime whose least prime factor is at least the gap between those primes.
Asks whether the largest prime divisor of n choose k is always at least the smaller of n minus k plus 1 and k to a power greater than one.
Splits n choose k into the part made of primes at most k and the part made of primes above k, and asks how the sizes of the two parts compare.
Asks whether, for k in a middle range, the distinct prime divisors of n choose k number about k times the sum of one over p for primes p between k and n.
Asks whether every integer at least 2 is a ratio of two products of k consecutive integers, for some k at least 2 with the blocks disjoint.
Asks for the order of the longest initial interval that one residue class per prime up to x can cover, the covering form of Jacobsthal's function; open between x log x times iterated logarithms and Iwaniec's x squared.
Asks for the largest exponent such that one residue class per prime between n to that exponent and n covers every integer from 1 to n, and whether it tends to zero; open, between Erdős's lower bound and a counting bound of 1/e.
Asks whether, for large n, one residue class per prime up to n can be chosen so that every integer from 1 to n lies in at least two of them; open on the site, with three pending AI-assisted full claims of April and July 2026.
The density of the integers whose kth smallest prime factor is a given prime p, and how that density behaves as k and p vary.
Asks for a necessary and sufficient condition on a set of positive integers for its set of multiples to have density one; open, with one family of block sequences settled by Tenenbaum.
Asks whether the density of integers with exactly one divisor in an interval from n to m is unimodular in m; disproved by Cambie, who also shows many local maxima.
Bounds the largest gap between consecutive integers above n having a divisor between n and twice n, asking if it is at most a power of the logarithm of n.
Estimates how large the ratio of the largest to the smallest integer with a given value of Euler's totient function can be for values up to x.
Asks how fast a chain of primes, each congruent to 1 modulo the previous one, must grow, and whether a nearly optimally slow such chain exists.
The longest chain of primes dividing n, and the longest chain of divisors of n, in which each term is congruent to 1 modulo the previous term.
Asks whether some threshold exponent splits the density of integers with a divisor above 1 congruent to 1 modulo m into a limit of zero below and one above.
Asks whether the greatest common divisor of n choose i and n choose j always tends to infinity with n, uniformly over all i and j between 2 and half of n.
Asks whether for all i less than j up to half of n some prime at least i divides both n choose i and n choose j.
Studies the least greatest common divisor of n and n choose k over k between 1 and half of n, asking when it is large and how big it can be for composite n.