Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
516 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Determines for which n and r the iterates of the map sending n to n plus Euler's totient of n satisfy that shifting by r steps eventually doubles the value.
Asks whether, for all m and n at least two, the iterated sum-of-divisors sequences starting at m and at n eventually share a value.
Asks whether infinitely many n have the property that every smaller m satisfies m plus its number of distinct prime factors being at most n.
Asks whether the iterates of the map sending n to n plus its number of divisors, started from any two integers, always eventually meet at a common value.
Asks whether the ratio of the count of totient values below x to the number of distinct totients of integers below x has a limit exceeding one; the limit's existence is open.
Asks whether the ratio of the divisor counts of the factorials of n plus a power of the logarithm of n and of n tends to infinity for large exponents.
Determines the behavior of a self-referential recursion whose terms are sums of earlier terms, and whether it misses infinitely many integers.
Estimates the growth of the sequence beginning one, two in which each term is the least larger integer that is a sum of consecutive earlier terms.
Estimates the largest subset of the first n integers with distinct pairwise products, and bounds sets whose products of r increasing members are distinct.
Asks for a set with positive relative density among the primes such that, for infinitely many n, subtracting each member from n always gives a prime.
Concerns the greedy decreasing sequence in which each term is the largest smaller integer above 1 whose prime factors all exceed n minus that term, and asks whether some term is composite for every large n.
Determines how dense the sumset of two infinite sets of natural numbers can be when all its elements are pairwise coprime.
Asks whether the limiting least start of m consecutive k-th power residues modulo p is finite for m equal to two, and for m equal to three with k odd.
Asks whether, for any exponent above one half and any large prime, every interval of that length contains two numbers whose product is one modulo the prime.
How long an interval must be so that at most a small fraction of its integers have a divisor strictly between n and twice n.
Estimates the least integer above twice k for which the product of its k preceding integers has no prime factor between k and twice k; claimed in a 2026 preprint to grow faster than any power of k, and at most exponential.
The longest interval inside x to twice x on which every integer has more than log log n distinct prime factors.
Asks whether the least sum of a symmetric pair of primes around the nth prime exceeds twice the nth prime by an unbounded amount infinitely often.
Asks whether a sequence of primes with non-decreasing gaps must have its nth term grow faster than n squared.
Asks whether the least common multiple up to one below the next prime is always less than the previous prime times the least common multiple up to it.
Asks whether the reciprocal sum of the terms below n of the greedy sequence making n minus each term pairwise coprime tends to infinity, and likewise for two subsequences.
Asks whether the number of distinct smooth parts, using primes below t, of the t consecutive integers after n is always at least a constant times t.
Asks whether the sum of the least prime factor over n, over a short interval near x, is always bounded below by a constant for large x.
Asks whether some function tending to infinity admits, for every large n, a composite number above n plus that function but below n plus its least prime factor.
Asks whether, for large x, residues can be chosen for the primes up to x and those primes split in two so every integer below x is covered by both parts; open, with one sentence of the 1980 monograph as its only source.