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 sum over n of the nth prime 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.
Estimates the largest lower bound for the maximum modulus on the unit circle of a product of terms one minus z to the a-i, over all choices of n exponents.
Asks whether the sum of one over two to the n minus one, taken over any infinite set of naturals, is irrational.
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 whether the sum of a-n divided by two to the a-n is irrational for every increasing sequence whose ratio to n tends to infinity.
Asks for which n the value n over two to the n is a sum of distinct terms k over two to the k, and whether some rational has uncountably many such sums.
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 two to the two to the n keeps reciprocal sums irrational under all asymptotically equal replacements, and whether such sequences must grow fast.
The powers-of-two case is false; asks whether factorial denominators keep their reciprocal sum irrational under every bounded nonzero integer perturbation.
Asks how fast an increasing integer sequence can grow when the sum of reciprocals of its terms and of its terms minus one are both rational.
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 sum of reciprocals of Fibonacci numbers along any geometrically growing index sequence must be 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 of reciprocals of the least common multiples of the first n integers built from a fixed finite set of primes is irrational.
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.
Determines explicitly, and bounds the growth of, the greedy sequence starting at 0 and n that avoids any three-term arithmetic progression.
The largest number of subsets of {1,...,N} with every pairwise intersection a non-empty arithmetic progression; open for the exact value, while Szabó's linear-error question, N^2/2 + O(N), has a Lean proof Conjectures.io accepted.
Asks whether there is a covering system of congruences whose moduli are all of the form p minus one for primes p at least 5.
Asks whether a group can be exactly covered by more than one coset when the cosets have different sizes, each element lying in exactly one.
Asks whether a system of r congruences that covers two to the r consecutive integers must cover every integer.