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 n vertices of a convex polyhedron in space always determine nearly half of n distinct distances.
Asks whether two sets of n planar points can have fewer than n over the square root of the logarithm of n distinct distances between the two sets.
Asks whether the least prime not dividing the product of k consecutive integers above n is at most about the logarithm of n, for fixed k and large n.
Concerns pairwise balanced designs on the first n integers, families of sets in which every pair of distinct elements lies in exactly one set.
Asks whether the exponent governing the largest clique forced when every p vertices span at least q edges is strictly increasing in q; open, with the 1997 source's endpoint bounds and a disputed upper bound at the top.
Asks whether the number of incongruent n-point planar sets maximizing the number of unit distances tends to infinity, and exceeds one for every n above three.
Bounds how many lines can pass through at least k, or through exactly k, of n given points in the plane.
Asks whether n points in d-dimensional space whose pairwise distances all differ by at least one must have diameter at least (1 + o(1)) n squared.
Asks whether a product of at least four positive terms in a primitive arithmetic progression, with gcd of initial term and difference one, can be a perfect power.
Asks which sets of integers are locally periodic, in that membership up to n is unchanged by some shift, for sums of two squares and other examples.
Asks whether every sufficiently large integer has the form a times a prime squared plus b, with a at least one and b less than that prime.
Asks whether two disjoint blocks of k consecutive integers can have the same least common multiple; Erdős's 1979 conjecture that they cannot is open, with a few solutions known only when the block lengths differ.
Asks whether infinitely many n have every n minus k with fewer distinct prime factors than about the logarithm of k over its own logarithm.
Asks whether every sufficiently large n admits some k for which the least prime factor of n plus k exceeds k squared plus one.
Asks whether every large n has some k for which n plus k is composite and the least prime factor of n plus k exceeds k squared.
Asks whether the largest prime divisor of n choose k is always at least the smaller of n minus k plus 1 and k to a power greater than one.
Splits n choose k into the part made of primes at most k and the part made of primes above k, and asks how the sizes of the two parts compare.
Asks whether, for k in a middle range, the distinct prime divisors of n choose k number about k times the sum of one over p for primes p between k and n.
Asks whether every integer at least 2 is a ratio of two products of k consecutive integers, for some k at least 2 with the blocks disjoint.
Asks for the order of the longest initial interval that one residue class per prime up to x can cover, the covering form of Jacobsthal's function; open between x log x times iterated logarithms and Iwaniec's x squared.
Asks for the largest exponent such that one residue class per prime between n to that exponent and n covers every integer from 1 to n, and whether it tends to zero; open, between Erdős's lower bound and a counting bound of 1/e.
Asks for a necessary and sufficient condition on a set of positive integers for its set of multiples to have density one; open, with one family of block sequences settled by Tenenbaum.
Bounds the largest gap between consecutive integers above n having a divisor between n and twice n, asking if it is at most a power of the logarithm of n.
Asks how fast a chain of primes, each congruent to 1 modulo the previous one, must grow, and whether a nearly optimally slow such chain exists.
Asks whether for all i less than j up to half of n some prime at least i divides both n choose i and n choose j.