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 greatest prime factor of the product of an irreducible polynomial's values up to n, in particular whether it exceeds n to a power above one.
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, for each k at least two, the number of ways to write n as a sum of k kth powers of primes is unbounded.
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 n points in the plane in convex position always include a vertex with at least half of n distinct distances to the other vertices.
The least r such that every k-element subset of the first n integers contains more than r members divisible only by primes from some set of r primes.
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 every prime p > 2 has a primitive root modulo p that is itself a prime smaller than p; the site's wording also includes p = 2, where it fails.
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.
Asks whether, for any increasing integer sequence, the discrepancy of its multiples of alpha stays near the square root of N for almost every alpha.
Asks whether the sequence counting the independent sets of each size in a tree or forest is unimodal.
Asks whether, for almost every alpha, the fractional parts of its integer multiples visit every measurable set in the unit interval with frequency its measure.
Estimates the growth of the sums of a square integrable function at the fractional parts of alpha times a lacunary integer sequence, for almost every alpha.
Asks how fast a square integrable function's Fourier partial sums must converge for its averages along alpha times a lacunary sequence to equal its integral.
Asks whether the fractional parts of alpha times the primes fail to be well distributed for every alpha.
Asks whether an interval with bounded discrepancy along the fractional multiples of an irrational alpha must have length a fractional multiple of alpha; the site's wording, asking that of both endpoints, is false.
Asks whether almost-all approximability of alpha by fractions within f of q over q equals divergence of the sum of the totient of q times f of q over q.
Concerns how many fractions with denominator the kth term of an integer sequence reduce to a denominator not equal to an earlier term of the sequence.