Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether every set of integers of density zero lies in the sumset of some set whose counting function is smaller than the square root of N.
Asks whether every additive basis of density zero has its sumset counting function grow infinitely faster than its own counting function.
Asks whether the integers that are sums of exactly r distinct elements of a basis of order r must have positive lower density.
Asks whether the sequence extending a finite set by the least integer that is not a sum of two earlier terms has eventually periodic differences.
Asks, after Folkman, whether a multiset of integers with linear counting function must have subset sums containing an infinite arithmetic progression, in the form Szemerédi and Vu prove, with one absolute constant; Folkman's question for every constant is open.
Asks whether every set of integers with at least a sufficiently large constant times the square root of N elements up to N, for every N, has subset sums containing an infinite arithmetic progression.
Asks whether a sequence complete after any finite deletion but never after an infinite one, with ratios bounded away from one, must have ratios tending to the golden ratio.
Asks whether some integer sequence with successive ratios tending to two has subset sums of density one even after any finite set of terms is removed.
Asks whether a finite set of integers with all subset sums distinct must have its reciprocals summing to less than two.
Asks whether the numbers p of n plus one over n, for a rational polynomial p with positive leading coefficient, stay complete after any finite set is removed.
Asks whether a planar measurable set of infinite measure must contain the vertices of an isosceles trapezoid of area one, or of other prescribed shapes.
Asks whether some sequence growing at least geometrically has finite sums of reciprocals of its terms covering every rational in some open interval.
Asks whether some positive c gives, for all large n, integers up to n whose sums over blocks of consecutive terms take at least c times n squared values.
Asks whether some infinite sequence of integers writes every large n as a sum of consecutive terms at least twice, or in ways tending to infinity.
The growth rate of the fewest classes needed to partition the numbers below n so that n is never a sum of distinct members of one class.
Bounds the number of subsets of an N-element set of naturals summing to a fixed target by 2^N over N^{3/2}, proved by Sárközy and Szemerédi with Stanley's exact maximizers, and the count with the subset size also fixed by 2^N over N squared, proved by Halász from his bound on signed sums of 1-separated plane vectors in a unit ball.
Asks whether only finitely many families of disjoint integer intervals, each of length at least four, have the product of all their members equal to a square.
Asks whether every large n has k consecutive n^epsilon-smooth integers up to n; trivially true as worded, it is proved in both nontrivial readings, each member smooth to its own power or the run inside [n/2, n].
Asks whether infinitely many integers n have both n and n plus one with largest prime factor below their own square root.
Asks whether the integers n whose largest prime factor is smaller than that of n plus one have density one half.
Asks whether infinitely many integers n have the largest prime factors of n, n plus one, and n plus two in strictly decreasing order.
Asks whether the integers n with at least r squarefree binomial coefficients in row n have a density, and whether that density is positive; both answered yes by Granville and Ramaré's 1996 theorem.
Asks whether the largest exponent S such that every binomial coefficient in row n is divisible by some prime to the power S is unbounded as n varies; answered yes in 2025 by Cambie, Kovač and Tao.
Asks whether the integers up to x lying in an interval whose product has a repeated largest prime factor are asymptotically as many as the n up to x divisible by the square of their own largest prime factor.
Asks whether the number of highly composite numbers up to x grows faster than any fixed power of the logarithm of x.