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.
Estimates the least integer above twice k for which the product of its k preceding integers has no prime factor between k and twice k; claimed in a 2026 preprint to grow faster than any power of k, and at most exponential.
The longest interval inside x to twice x on which every integer has more than log log n distinct prime factors.
Asks whether, for all large n, some symmetric pair of primes around the nth prime has product exceeding the square of the nth prime.
Asks whether the least sum of a symmetric pair of primes around the nth prime exceeds twice the nth prime by an unbounded amount infinitely often.
Asks whether a sequence of primes with non-decreasing gaps must have its nth term grow faster than n squared.
Compares the least prime congruent to one modulo n with the least integer whose Euler totient is divisible by n; a September 2026 forum claim answers the three questions no, no and yes, pending review.
Whether some epsilon > 0 gives infinitely many n with every prime up to (2 + epsilon) log n dividing the product of the next log n integers: yes, by a 2026 AI construction the site accepted; Erdős's opposite 1979 form: no.
Asks whether the least common multiple up to one below the next prime is always less than the previous prime times the least common multiple up to it.
Estimates the largest v such that no integer strictly between u and v is built only from primes dividing the product of u and v.
Asks whether the reciprocal sum of the terms below n of the greedy sequence making n minus each term pairwise coprime tends to infinity, and likewise for two subsequences.
Asks whether the number of distinct smooth parts, using primes below t, of the t consecutive integers after n is always at least a constant times t.
Asks whether the sum of the least prime factor over n, over a short interval near x, is always bounded below by a constant for large x.
Asks whether some function tending to infinity admits, for every large n, a composite number above n plus that function but below n plus its least prime factor.
Asks whether every lacunary sequence admits an irrational multiplier whose fractional parts along the sequence are not dense in the unit interval; proved by Pollington and de Mathan, with Peres and Schlag's separation of order epsilon over log(1/epsilon), while the site's wording is trivially true.
Asks whether the largest set of points in a disc of radius X whose pairwise distances all stay at least delta from the integers has o(X) points, and even fewer than X to the one half plus o(1); proved by Sárközy (the first bound) and Konyagin (the sharp exponent one half).
Asks whether for some fixed delta the largest set of points in a disc of radius X whose pairwise distances all stay at least delta from the integers grows without bound as X grows; the corrected statement takes the limit in X, which the site misprints, and Sárközy's power lower bound proves it.
Asks whether, for large x, residues can be chosen for the primes up to x and those primes split in two so every integer below x is covered by both parts; open, with one sentence of the 1980 monograph as its only source.
Studies the set of partial sums of the increasing divisors of n above one.
Asks whether the reciprocal sum converges over integers that are sums of distinct proper divisors of themselves while no proper divisor has that property.
Studies weird numbers, those whose divisor sum is at least twice the number yet which are not the sum of any set of their own divisors.
Asks whether some finite starting set of primes grows without bound under repeated adjunction of all primes that are sums of three distinct members; yes, by Vinogradov's three-primes theorem, an observation the site accepted.
Asks whether some starting list of primes makes infinite the sequence whose next term is the least prime of the form previous plus an earlier term minus one.
Asks whether the positive integers can be permuted so that every two consecutive terms sum to a prime; yes, by an unpublished construction of Odlyzko reported by Erdős and Graham in 1980 and accepted by the site.
Which set theoretic assumptions allow a three-coloring of the plane in which every uncountable set contains a pair of points of each color.
Whether every finite set of nonzero residues modulo a prime can be ordered with all partial sums distinct (Graham's rearrangement conjecture); proved for all large primes by four range results with no explicit threshold.