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 infinitely many central binomial coefficients are coprime to one hundred and five.
Asks whether the sum of the reciprocals of the primes up to n that do not divide the central binomial coefficient of n is bounded by a constant.
Asks whether the integers n with at least r squarefree binomial coefficients in row n have a density, and whether that density is positive; both answered yes by Granville and Ramaré's 1996 theorem.
Asks whether the largest exponent S such that every binomial coefficient in row n is divisible by some prime to the power S is unbounded as n varies; answered yes in 2025 by Cambie, Kovač and Tao.
Asks whether the integers up to x lying in an interval whose product has a repeated largest prime factor are asymptotically as many as the n up to x divisible by the square of their own largest prime factor.
Asks whether the number of highly composite numbers up to x grows faster than any fixed power of the logarithm of x.
Asks how long an interval of integers can be when the largest prime dividing its product appears at least twice, and whether that length can be arbitrarily large.
Asks whether, for every k, infinitely many primes p are the largest prime factor of the product of p squared through p squared plus k.
Asks whether, for 1 < k < n - 1, every binomial coefficient n choose k has a prime divisor at most n/2, except 7 choose 3; proved by Ecklund in 1969, while the site's strict bound p < n/2 fails at 4 choose 2.
Asks whether the largest value of a composite number below n plus its least prime factor exceeds n for all large n, and whether the excess grows without bound.
Asks whether a binomial coefficient, other than the trivial ones, can be a product of consecutive primes infinitely often.
Asks whether some constant c makes every binomial coefficient in row n have a divisor between c times n and n.
Asks whether two disjoint blocks of more than three consecutive integers can have equal products only finitely often, and whether such cases can be classified.
Asks whether every positive integer n admits some k for which the product of the first k integers from n divides the product of the next k.
Asks whether the least top factor in a factorization of n factorial into increasing factors above n exceeds two n by about a constant times n over log n.
Bounds the largest possible smallest factor when n factorial is written as a product of n increasing factors, in particular whether it approaches n over e.
Asks whether the fewest factors needed to write n factorial as increasing factors of size at most n squared is about n over two minus n over two log n.
Determines the behavior of the smallest spread between largest and smallest factor when a factorial is written as a product of distinct increasing integers.
Bounds the average least starting point m for which n divides a product of k consecutive integers from m, and asks whether these averages shrink as k grows.
Asks whether random signs on n unit complex numbers give a sum of absolute value at most the square root of two with probability at least about one over n; Erdős asked it with radius one, which fails for every even n.
Asks whether, for every k, some n makes the product of the k plus one integers from n minus k to n divide the central binomial coefficient of n.
Asks whether only finitely many equalities hold between two products of central binomial coefficients taken over distinct indices.
Asks whether a factorial is one less than a perfect square only for n equal to four, five, and seven.
Asks whether a factorial can equal a sum or difference of two kth powers with k greater than two and the powers not both trivial.
Asks for the average and typical size of the largest excess of a sum of numbers whose factorials divide n factorial over n, for each fixed count k of terms.