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 a function additive for almost all pairs of reals must agree almost everywhere with a function that is additive for all pairs.
Asks whether real n-dimensional space splits into countably many sets in each of which all pairwise distances are distinct.
Asks whether every two-coloring of a product of three sets of size aleph one contains a monochromatic product of three countably infinite subsets.
Asks a question about how large the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one can be.
Asks for the extreme behavior of sums of the fundamental Lagrange interpolation polynomials built from nodes in the interval from minus one to one.
Asks for the least value, over n nodes in the interval from minus one to one, of the integral of the sum of squares of the Lagrange basis polynomials, and whether that least value is 2 minus (1+o(1))/n.
Asks for the best possible bounds on the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one.
Asks whether, for each positive constant, a small positive number exists making a stated property hold for all large degrees of polynomial interpolation.
Asks whether the smallest set containing one and closed under tripling plus one, doubling plus one, and sextupling plus one has positive lower density.
Asks whether every orbit of the shortcut Collatz map reaches one; open, with the site's caveat on the reported Erdős prize figure, verification below 2^71, no cycle with at most 91 local minima, and Tao's almost-all theorem.
Asks whether some set of naturals of lower density above one third has no two members, possibly equal, summing to a power of two.
Asks whether the largest product of two consecutive prime gaps below x is negligible compared with the square of the largest prime gap below x.
Asks whether an interval of length a constant times the maximal prime gap below x contains the expected number of primes, for y between x halved and x.
Asks whether the gaps in the sequence of integers with at most two prime factors are infinitely often much larger than the logarithm of the index.
Asks whether infinitely many n make n - 2x^2 prime for every x with 2x^2 < n.
Asks whether infinitely many n have n minus k squared prime for every k coprime to n with k squared below n.
Asks whether infinitely many n, or any n above one hundred and five, make n minus every power of two greater than one and less than n prime.
Estimates how many multiples of at least one of finitely many given primes every interval of k consecutive positive integers must contain.
Asks whether a random completely multiplicative sign function almost surely has partial sums up to N exceeding any multiple of the square root of N.
Asks a question about two increasing sets of positive integers whose nth elements have ratio tending to one.
Asks a question about essential components, sets whose sum with any other set of Schnirelmann density strictly between zero and one raises that density.
Asks whether the set of n with alpha times n squared within one over the logarithm of n of an integer is an additive basis of order two, for irrational alpha.
Asks whether every large integer is a sum of two squares minus a square with all three squares at most that integer.
Determines the density of integers n whose greatest common divisor with the integer part of n to the power alpha is one, for non-integer positive alpha.
Asks whether every plus or minus one polynomial of degree n has maximum modulus on the unit circle above (1+c) times the square root of n for some fixed c>0; answered no by the OpenAI release's construction of 2026-09-23.