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.
Estimates the largest possible upper density of a measurable planar set containing no two points at distance one, and asks whether it is at most one quarter.
Asks whether the distribution of normalized gaps between integers coprime to the product of the first k primes tends to a continuous limiting function.
Asks whether any set of integers with at least logarithmically many elements up to N gives some integers unboundedly many representations as prime plus member.
Asks whether every multiplicative function taking only the values plus and minus one has a mean value.
Asks whether some infinite set of primes has the property that the gaps between consecutive integers built only from those primes tend to infinity.
Asks whether a sparse infinite set of naturals must have its sumset, counted up to N, at least three times as large as the set, up to o(1), along a sequence of N.
Asks whether, for coprime a and b, every large integer is a sum of distinct numbers of the form a to the k times b to the l.
Asks whether there are infinitely many n for which the number of distinct prime factors of n plus k stays of order k for every positive k.
Asks whether the sum over n of the sum-of-divisors function of n divided by two to the n is irrational.
Asks whether, for every positive k, the sum over n of the sum of kth powers of the divisors of n divided by n factorial is irrational; answered yes by a 2026 kernel-checked Lean proof rebuilt, replayed and statement-audited here.
Asks whether a sequence with ratios tending to one whose distinct subset sums meet every arithmetic progression infinitely often represents all large integers.
Asks whether a set that grows in every dyadic range and has divergent sums of distances to the nearest integer for every angle represents all large integers as distinct sums.
Asks whether every infinite sequence in the unit interval has some subinterval whose counting discrepancy is unbounded.
States that the divisor-function product-denominator series is irrational for every positive-integer sequence tending to infinity.
Asks whether the sum over squarefree n of n divided by two to the n is irrational.
Asks how slowly an increasing integer sequence can grow while every sum of reciprocals of positive integer multiples of its terms stays irrational.
Asks whether every sequence of positive integers with convergent reciprocal sum admits a positive integer shift making the shifted reciprocal sum irrational.
Asks whether the triples of reciprocal sums over n, n plus one and n plus two, taken over infinite sets with convergent reciprocal sum, contain an open set.
Asks whether the sum over n of one over the product of the f of n consecutive integers starting at n plus one is irrational whenever f tends to infinity.
Asks whether a system of r congruences that covers two to the r consecutive integers must cover every integer.
Asks whether, for every c, some n has sum of divisors above c times n yet admits no covering system whose moduli are distinct divisors of n above one.
Asks whether a sequence of moduli growing faster than k log k always leaves a number of uncovered integers below each modulus that is not small compared with k.
Asks whether moduli whose congruences always leave a density-zero set must have a finite initial segment leaving density below any given epsilon.
Asks whether, for a suitable polynomial, every large integer is the sum of its values over the denominators of some unit fraction representation of one.
Asks whether the largest possible smallest denominator among k distinct unit fractions summing to one is asymptotically k divided by e minus one.