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.
Asks whether the number of ways of writing n as a sum of two powerful numbers is smaller than any fixed power of n.
Estimates the longest run of consecutive integers below x with all divisor counts distinct, and whether short intervals must repeat a divisor count.
Asks whether the complement of any sum-free set of reals contains a set of size continuum whose pairwise sums also avoid it; open; AlphaProof's Lean proof of the Sidon case is a pending claim and a thread comment argues the Baire case.
Asks for the limit inferior, limit superior and growth of the sum of the reciprocals of n minus p taken over all primes p below n.
Asks whether a sequence of reals whose distinct integer power products always differ by at least 1 has no more terms up to x than there are primes up to x; the quantifier over x is implicit, and the two readings are parts.
The largest possible measure of a set inside a disc of radius r containing no two points at an integer distance apart.
Concerns the growth of the sequence starting 0 and 1 in which each new term is the least n for which the number of pairwise sums at most n is less than n.
Asks whether every density-zero set has a density-zero preimage under the sum of proper divisors; known results cover the primes, sums of two squares, prime-factor-count tails, palindromes, sparse and missing-digit targets.
Estimates the largest possible gap between the two highest distance multiplicities determined by a set of n points in the plane.
Estimates the least n such that every set of n consecutive integers above k contains one divisible by a prime greater than k; open between a Rankin-type prime-gap bound and k over log k times iterated logarithms.
Estimates the greatest k for which some run of k consecutive integers starting at most at n has every term divisible by a prime larger than k; log k(n) is between c sqrt(log n log log n) and (log n)/2; both questions open.
Estimates the largest dissociated subset guaranteed in any set of n reals, in particular whether it always has at least floor(log_2 n) elements.
Determines the order of magnitude of the error term when the count of squarefree integers up to x is compared with six over pi squared times x.
Asks for the order of magnitude of Jacobsthal's function over integers with at most k distinct prime factors, and whether it is O(k^2); the quadratic bound is proved by an accepted partial claim, and the order of magnitude stays open.
Asks whether, for irrational alpha > 1, infinitely many primes p have the integer part of p alpha prime; open, the one-prime statement classical and the two-prime statement known only for almost all alpha.
Asks whether the sum of the number of divisors of the values of an irreducible integer polynomial up to X is asymptotic to a constant times X log X.
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, 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 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 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.
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 whether Euler's totient function takes the same value at n and at n plus one for infinitely many n.
Asks whether, for every fixed positive c and all large x, some n up to x has all totient values on an interval of length a power of the logarithm of x distinct.
Determines the least number of edges forcing a triangle in a graph on n vertices whose chromatic number is at least r; known exactly for r up to three, for r equal to four and n large, and open in general.