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, for infinitely many n, two numbers whose factorials' product divides n! times the n-th power of the product of the first r primes can sum to more than n plus f(r) log n, with f(r) tending to infinity.
Asks for a proof that every finite set of integers has two members whose greatest common divisor is at most one member divided by the set's size.
Asks whether a power of two can equal a sum of distinct factorials in only finitely many ways.
Asks for which integers a and primes p the power of p dividing a sum of increasing factorials starting at a is bounded, and how that bound behaves.
Asks whether, for each odd prime p, the equation with p minus one factorial plus a power of a equal to a power of p has only finitely many solutions.
Asks whether only finitely many powers of two are written with just the digits zero and one in base three.
Asks whether the number of ways to write an integer as a power of two plus a power of three plus a product of a power of two and a power of three is bounded.
Asks whether the number of totient iterations needed to reach one, divided by the logarithm of n, has a distribution or is almost always constant.
Asks how many iterations of the map sending n to Euler's totient of n plus one are needed to reach a prime, and how often a given prime is reached.
Asks whether the k-th iterate of the sum-of-divisors function, taken to the power one over k, tends to infinity for every integer n at least two.
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.
Estimates the largest k such that every ordering pattern of k consecutive values of Euler's totient function occurs below n, and which pattern fails first.
Asks whether the count of totient values up to x doubles when x doubles, and whether that count has an asymptotic formula; the doubling limit is proved twice in Lean, and the release's asymptotic equivalent is a pending claim.
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 infinitely many positive integers are not of the form n minus Euler's totient of n.
The set of limit points of the ratio of the number of divisors of n plus one factorial to the number of divisors of n factorial.
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.
Asks whether there is an increasing sequence of density one all of whose products of consecutive blocks of terms are distinct; answered yes in July 2026 by Chojecki and Sneiderman, accepted by the site, not refereed.
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.
Asks whether the integers eventually produced from two and three by repeatedly adjoining products of two distinct terms minus one have positive density, read as positive lower density following the site's curator.
Estimates the largest subset of the first n integers with distinct pairwise products, and bounds sets whose products of r increasing members are distinct.