Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether for every k at least 4 and r at least 1 some k-chromatic graph has every vertex critical and no critical set of at most r edges; settled by a Lean proof the bounty site Conjectures.io certified in September 2026.
Asks whether there are infinitely many n for which n and n plus 1 have the same number of divisors.
Asks whether an exact covering system exists, finitely many congruence classes with distinct moduli such that every integer satisfies exactly one of them.
Asks whether some bound and some number of colors force, in every coloring of the integers, a slowly growing sequence whose subset sums miss a color; no, by a 2026 AI-generated coloring accepted by the site after review.
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.
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.
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.
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.
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 greatest prime factor of two to the n minus one, divided by n, tends to infinity.
Concerns the values of an irreducible integer polynomial with positive leading coefficient whose degree exceeds two and is not a power of two.
Asks whether the sum over primes below x of the least kth power nonresidue is asymptotic to a constant times x over log x; proved for every k under Elliott's convention (Erdős 1961 for k=2, Elliott 1967); the sum over every prime with a kth power nonresidue is an open variant for composite k.
Asks whether the sum over primes below x of the eventual-time threshold of the Legendre-symbol partial sums is roughly x over log x; proved by Elliott (1969) for the two-sided threshold, the one-sided form left to his remark.
Asks whether the naturals can be two-colored so that every monochromatic arithmetic progression starting at a has fewer terms than any fixed power of a.
Asks whether, for fixed s at least three, the Ramsey number of s against k is at least k to the s minus one over a power of log k.
Concerns the limiting sizes of the exponential sums of an infinite sequence in the unit interval taken at integer frequencies.
Concerns how small the spherical cap discrepancy of a finite set of points on the unit sphere can be made.
Concerns how slowly the discrepancy of an infinite plane sequence, measured against circles of radius r and their area, can grow.
Asks whether a polynomial's root arguments are equidistributed with error at most the square root of the number of nonzero coefficients times a log factor.
Concerns the distribution of the point sets on the unit sphere that maximize the product of all pairwise distances.