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 the set of sums of distinct pairs from a subset of the integers modulo a prime has size at least twice the subset size minus three, or the prime (the Erdős–Heilbronn conjecture); proved in 1994.
Asks whether the image of an integer polynomial of degree at least two admits a unique additive complement in the integers.
Asks whether the number of distinct factorial residues modulo a prime is asymptotically one minus one over e times the prime.
Asks whether, for every k other than one, there are infinitely many n with two to the n congruent to k modulo n.
Asks whether every sequence in [0,1] has a gap n for which the lower limit of n times the spacing of terms n apart is at most one over root five; proved by Chung and Graham with the sharp constant 0.3944....
Asks whether iterating a family of affine maps from the value one must repeat an element when the reciprocals of the multipliers sum to over one; yes, by Klarner's 1982 theorem, its 2022 extension and a 2025 thread proof.
Asks for analogs, for root m and other algebraic numbers, of the Graham-Pollak recurrence whose differences a_{2n+1} - 2a_{2n-1} are the binary digits of root two; solved by Stoll's families for every positive real and every base.
Estimates the least N forcing a monochromatic solution of a plus b equals c in every k-coloring of one through N, and asks whether it is exponential in k; open between c times 3.28 to the k and (e minus 1/6) times k factorial.
Asks whether every k-coloring of the first N integers leaves a positive proportion of them expressible as a sum of two distinct integers of one color.
Asks whether the least possible number of nonzero terms in the square of a rational polynomial with exactly k nonzero terms tends to infinity as k grows.
Asks whether the set of integers avoiding a prescribed residue class pattern modulo each member of a given set of moduli always has a logarithmic density.
Asks whether every set of integers of positive density contains three distinct members one of which is the least common multiple of the other two; true by Kleitman's union-free theorem, attested here second-hand.
Asks whether the density of the multiples of a finite set up to m is under twice its density up to a smaller n at least the largest element; a counterexample claim of September 2026 is pending and unreviewed.
Asks whether the average of the squared gaps between consecutive integers divisible by no member of a sparse set of divisors tends to a finite limit.
Asks whether two subsets of the first N integers with all pairwise products distinct must have size product at most about N squared over the logarithm of N; proved by Szemerédi (1976), with a second proof by Erdős and Szemerédi.
Asks whether an additive function whose consecutive differences stay bounded must equal a constant multiple of the logarithm plus a bounded error.
Asks whether, for a real sequence tending to infinity whose consecutive ratios tend to one, the positions of the multiples of almost every real within the sequence's gaps are uniformly distributed; LeVeque's question, disproved by Schmidt. The site's wording restricts to integer sequences, for which the Davenport–Erdős theorem gives yes.
Asks whether there is a fixed k such that every large enough integer is the product of k integers at least two minus their sum.
Asks whether, for k greater than two, the multiset of all sums of k distinct elements of a finite set of complex numbers determines the set, given its size.
Asks whether, for all pairs of real numbers, the product of n with the distances from n times each number to the nearest integer has limit inferior zero.
Asks whether, for every positive irrational alpha, sums of two positive squares come arbitrarily close to alpha times a positive square; the site's wording over every irrational alpha fails at negative ones.
Determines the number of antichains of subsets of an n-element set, that is, families in which no member contains another.
Asks whether at most the middle binomial coefficient of the signed sums of n complex numbers of modulus at least one can lie in one open unit disc.
Asks whether every n by n doubly stochastic matrix has a permutation along which the product of entries is at least n to the power minus n.
Asks for the largest number of triples on n vertices with no four vertices carrying all four of their triples, Turán's tetrahedron problem; the density lies between Turán's 5/9 and a flag-algebra bound of 0.5615.