Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
99 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
The least area of the region where a monic polynomial with all roots from a fixed closed infinite set of complex numbers has absolute value below one.
Examines a polynomial whose roots all lie strictly inside the unit disc and the shape of the region where its absolute value is below one.
Studies the least m such that n is the sum of the k smallest divisors of m for some k, and how that function behaves.
Asks whether infinitely many primes p have p minus k factorial composite for every k with k factorial less than p.
Counts pairs with the sum of divisors of a plus that of b equal to that of a plus b and a plus b at most x, and asks whether the count is proportional to x; Li's June 2026 preprint claims it exceeds x times any log power, pending.
Concerns a constant larger than r to the power minus r, for each r at least three, in a property of r-uniform hypergraphs on many vertices.
Estimates the least number of distinct distances determined by n points in d-dimensional space, asking whether it is nearly n to the power two over d.
Concerns the number of points on the circle of radius r at which a non-monomial entire function attains its maximum modulus.
Asks whether an additive function that decreases from n to n plus one only for a density-zero set of n must be a constant multiple of the logarithm.
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 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 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 the Lagrange interpolation polynomial of degree n minus one matching a function at n given nodes in the interval from minus one to one.
Asks a question about fixed sets of n distinct interpolation nodes in the interval from minus one to one together with a tolerance tending to zero.
Asks whether the number of sites ever favorite by time n in planar simple random walk is eventually bounded by a power of log n almost surely.
Concerns the families of three-uniform hypergraphs of a given chromatic number that avoid a fixed finite three-uniform hypergraph.
Estimates the largest edge counts for connected graphs on n vertices whose Ramsey number against a triangle equals two n minus one; a 1996 preprint claims a linear threshold for all such graphs, a no to the closing question.
Asks whether every coloring of the integers with finitely many colors contains k primes in arithmetic progression all of the same color.
Concerns covering systems with distinct moduli that are minimal, in that no proper subsystem still covers every integer.
Concerns sets of distinct moduli that can cover the integers by some choice of residues but have no proper subset that can.
Asks whether every two-coloring of the natural numbers admits an infinite set whose pairwise sums, doubles included, share one color (Owings's question); open on the site, false for three colors, claimed for two.
Asks whether, for every epsilon and eta, some k makes the largest prime factor of n(n+1)...(n+k) exceed n to the one minus epsilon for a set of n of density at least 1 - eta, the density read as lower density following the site's curator.
The largest number of congruences guaranteed, over choices of one residue class modulo each n up to x, to be satisfied by every integer up to x; the site's own argument gives log x plus lower-order terms, a pending claim.
Asks for a path to infinity through coprime pairs above 1 with a composite coordinate, steps changing one coordinate by one; Erdős first asked it without the composite condition, which Stewart quickly answered yes.