Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
99 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the edges of any chordal graph on n vertices can be partitioned into about n squared over 6 cliques.
Asks whether every convex polygon has a vertex with no four other vertices at the same distance from it; Erdős first asked it with three, which Danzer's convex nonagon refutes.
Estimates how many Abelian subgroups are needed to cover a group in which every set of more than n elements contains two distinct commuting elements.
Asks for a bound of the form C^(sqrt n) on the least N forcing, in any r-coloring of K_N, n vertices missing some color's triangle; a random coloring refutes the site's wording, and no source gives another intended form.
Estimates the largest subset of the first N integers in which no element divides the sum of any distinct others; the displayed root-N question is answered no, and the order carries one unreviewed claim of exponent 1/5.
Asks whether a countable set of reals above one where every integer multiple of an element is at distance one or more from another must be sparse.
Asks whether every finite coloring of the positive integers admits arbitrarily large finite sets whose sums and products of distinct members share one color.
Asks whether the Ramsey number of the n-dimensional hypercube graph is at most a constant times its number of vertices; open on the site, with the linear bound claimed in full by a 2026 OpenAI release preprint.
Asks whether every integer greater than 2 has four over it written as a sum of three reciprocals of distinct positive integers.
Asks whether the sum of a-n divided by two to the a-n is irrational for every increasing sequence whose ratio to n tends to infinity.
Asks whether the sum of reciprocals of Fibonacci numbers along any geometrically growing index sequence must be irrational.
The maximum density of integers covered by choosing one congruence class for each modulus in a finite set, and whether equal classes minimize the density.
Asks whether, for every k at least 3, congruence classes can be chosen modulo each prime so that all large integers lie in one of them with quotient at least k.
Asks whether, for every large k, one can be written as the sum of reciprocals over k separated intervals of integers, each of length at least two.
Asks whether every positive rational with squarefree denominator is a sum of distinct unit fractions whose denominators are all products of two distinct primes.
Asks whether a minimal basis of order two exists whose kth smallest element divided by k squared tends to a nonzero constant; Erdős first asked it for any basis of order two, which Cassels answered yes.
Asks how large a subset of the first N integers can be if the sum of any two distinct members never divides their product, or never divides twice it.
Determines the limit of h of r divided by r squared, where h of r is the largest finite exact order of an additive basis of order r.
Determines for which m less than n a complete sequence can stay complete after removing any m elements yet fail after removing any n elements.
Determines how many integers up to n need exactly k factorials, with the largest being that integer's, to form a square product, for k from three to six.
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.
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.
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 a factorial is one less than a perfect square only for n equal to four, five, and seven.
Estimates the largest k such that every ordering pattern of k consecutive values of Euler's totient function occurs below n, and which pattern fails first.