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 a sequence of reals whose distinct integer power products always differ by at least 1 has no more terms up to x than there are primes up to x; the quantifier over x is implicit, and the two readings are parts.
Asks whether there is an infinite sequence of distinct Gaussian primes in which consecutive terms are always a bounded distance apart; the Gaussian moat problem, answered negatively by an accepted 2026 claim with a Lean proof.
The largest possible measure of a set inside a disc of radius r containing no two points at an integer distance apart.
Concerns the growth of the sequence starting 0 and 1 in which each new term is the least n for which the number of pairwise sums at most n is less than n.
Asks whether every density-zero set has a density-zero preimage under the sum of proper divisors; known results cover the primes, sums of two squares, prime-factor-count tails, palindromes, sparse and missing-digit targets.
Determines the largest number of unit distances among n disjoint translates of a compact convex set, in particular whether it exceeds n to a power above 1.
Asks whether the multiplicities of the smallest and largest distances among n points in the plane have product at most (9/8 + o(1)) n^2.
Asks whether n planar points with n-1 distances of multiplicities n-1, ..., 1 must be equally spaced on a line or a circle.
Estimates the largest possible gap between the two highest distance multiplicities determined by a set of n points in the plane.
Determines how many ordinary lines a set of n planar points with no k on a line must have to force r of the points to span only ordinary lines.
Estimates the least n such that every set of n consecutive integers above k contains one divisible by a prime greater than k; open between a Rankin-type prime-gap bound and k over log k times iterated logarithms.
Estimates the greatest k for which some run of k consecutive integers starting at most at n has every term divisible by a prime larger than k; log k(n) is between c sqrt(log n log log n) and (log n)/2; both questions open.
Estimates the largest dissociated subset guaranteed in any set of n reals, in particular whether it always has at least floor(log_2 n) elements.
Asks whether the ratios of the number of divisors of n plus one to the number of divisors of n are dense in the positive reals.
Asks whether every two-coloring of the reals admits a set of size aleph one whose sums of two distinct elements share one color; false in ZFC by Komjáth and by Soukup and Weiss, after a CH proof by Hindman, Leader and Strauss.
Asks whether, for k and r at least two, some set with no arithmetic progression of length k plus one has a monochromatic k-term one in every r-coloring; true, by Spencer's 1975 restricted van der Waerden theorem.
Asks whether, for integer sequences whose reciprocals sum finitely, one plus the sum of their reciprocals to the power one plus i t is never zero.
Asks whether the set of n for which the nth prime divided by n is less than the next such ratio has positive density.
Determines the order of magnitude of the error term when the count of squarefree integers up to x is compared with six over pi squared times x.
Asks for the order of magnitude of Jacobsthal's function over integers with at most k distinct prime factors, and whether it is O(k^2); the quadratic bound is proved by an accepted partial claim, and the order of magnitude stays open.
Asks whether the least prime congruent to a modulo d exceeds a fixed factor above Euler's totient of d times log d for many residues a; two 2026 proof claims answering yes, one with a Lean development, pending and unreviewed.
Asks whether, for irrational alpha > 1, infinitely many primes p have the integer part of p alpha prime; open, the one-prime statement classical and the two-prime statement known only for almost all alpha.
Asks whether n complex numbers of modulus at least one, the first equal to one, can keep all their power sums below an exponentially small bound.
Asks whether complex numbers whose power sums vanish on infinitely many blocks of n minus one consecutive indices are essentially the nth roots of unity, read as Tijdeman's classification.
Asks whether the sum of the number of divisors of the values of an irreducible integer polynomial up to X is asymptotic to a constant times X log X.