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 summed count of large distinct prime factors over k consecutive integers is infinitely often at most k, and about its extreme growth rate.
Asks whether, for each k at least 2 and every large enough n, the interval of length p_1...p_k starting at n contains an integer with more than k distinct prime factors.
Asks for a necessary and sufficient condition on an increasing sequence for a primitive sequence, no term dividing another, to grow no faster than it.
Asks whether the ratio of the summed divisor counts of two to the power k minus one, over k up to twice n and up to n, tends to a limit.
Estimates the least number of edges in an n-uniform hypergraph that cannot be colored with two colors but can with three.
Estimates the least order of a tournament in which every n vertices have a common dominator; open, with Erdős's 1963 upper bound and the Szekeres and Szekeres lower bound of 1965 a factor of order n apart, exact only to n = 3.
Asks whether every connected set in n-dimensional space has a connected subset that is neither a single point nor homeomorphic to the whole set.
Asks whether the size Ramsey number of every graph with n vertices and at least Cn edges exceeds the edge count by a factor growing faster than linearly in C; open, with no result found beyond Erdős's 1982 statement.
Estimates how many distinct exponents occur in the prime factorization of n factorial.
Asks whether infinitely many n have all exponents distinct in the prime factorization of n times n plus one.
Separates the proved quadratic lower bound, open k=6 case, and disproved nonmultiples-of-three part of the proposed general asymptotic.
Asks whether some graph has aleph two vertices and chromatic number aleph two while every subgraph on aleph one vertices is countably colorable.
Asks whether some graph on the ordinal omega two squared has chromatic number aleph two while every subgraph of smaller type is countably colorable.
Asks for the least prime cutoff x such that a positive density of blocks of k consecutive integers have every member divisible by a prime up to x; the inverse of Problem 687's covering function, open above the square root of k.
Asks whether, for every r, some k makes the product of all integers in any r disjoint intervals of length at least k never a perfect power.
Asks whether only finitely many disjoint blocks of consecutive integers, of lengths k1 and k2 at least 3, have products with the same prime factors.
Asks whether infinitely many prime gaps contain at least two integers all of whose prime factors are smaller than the length of that gap.
Asks whether the largest divisor of n times n plus 1 built only from the primes 2 and 3 exceeds any fixed multiple of n log n for suitable n.
The least number of edges forcing a graph of maximum degree at most d to have two edges at distance at least t; open, exact for t = 1, t = 2 and h_3(3) = 23, between 0.629^t d^t and 3d^t/2 + 1 in general, with 2026 preprints at t = 3.
Asks whether the powerful part of n(n+1)...(n+l) is below n^(2+eps) for every eps once n is large, whether its ratio to n squared is unbounded when l is at least 2, and whether its ratio to n^(l+1) tends to zero.
Asks whether 2 to the n plus or minus 1 and n factorial plus or minus 1 are powerful numbers for only finitely many n.
Concerns the increasing sequence of powerful numbers, those integers divisible by the square of every prime dividing them, and the gaps between them.
Concerns sums of coprime r-powerful numbers; the 3-powerful triple question is answered yes (Nitaj 1995), infinitely many solutions exist for every r at least 6, and r = 4 is open.
Concerns r-powerful numbers for r at least 3, the integers divisible by the r-th power of each of their prime factors.
Estimates how many powerful integers lie between consecutive squares, in particular whether some fixed power of log n bounds that count for every n and is nearly reached for infinitely many n.