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, for every constant C at least 0, the gap after the nth prime divided by log n tends to exactly C along some sequence of indices n.
Asks whether there is a covering system of congruences whose moduli are distinct and all odd.
Asks whether the odd integers that are not the sum of a prime and two powers of 2 have positive upper density.
Asks whether some fixed k makes every large integer the sum of a prime and at most k powers of 2; open, with three powers shown insufficient for infinitely many even integers.
Asks whether every large odd integer is the sum of a squarefree number and a power of 2.
Asks how large an infinite set with no element dividing the sum of two larger elements can be, in counting, density and reciprocal-sum terms; the first two questions have 2026 claimed answers, the reciprocal sum is open.
Asks whether the alternating sum of n divided by the nth prime converges; open, with Tao's proof of convergence conditional on a strong Hardy-Littlewood prime tuples conjecture.
Asks whether infinitely many primes p have every even number up to p minus 3 expressible as a difference of two primes not exceeding p.
Asks how few distinct divisors of a practical number represent every smaller integer, in particular for factorials; h(n!) < n^{o(1)} is proved (Conjectures.io, 2026), the (log log m)^{O(1)} part claimed, the rest open.
Asks whether a graph made of n edge-disjoint copies of the complete graph on n vertices has chromatic number exactly n.
Asks whether the number of n-element sets needed to force a k-sunflower grows only exponentially in n, with a base depending on k.
Asks whether every triangle-free graph on 5n vertices can be made bipartite by deleting at most n squared edges.
Asks whether the integers avoiding a chosen residue class for each modulus in an increasing sequence always have a logarithmic density.
Asks whether a set whose sumset omits only finitely many integers must have integers with arbitrarily many representations as sums of two elements.
Asks whether the largest Sidon set in the first N integers has size the square root of N plus an error smaller than every power of N.
Asks how sparse a set can be if every large integer is a prime plus one of its members, measured against the square of the logarithm of N.
Asks how sparse a set can be if every large integer is a square plus one of its members, measured against the square root of N.
Asks for the largest constant c such that every split of the first 2N integers into two equal halves has a difference realized at least c times N ways.
Asks whether an infinite Sidon set can contain nearly the square root of N elements up to N, for every positive tolerance.
Asks which growth rates just below the square root of N force some integers to have arbitrarily many representations as sums of two elements.
Asks whether an infinite set with all triple sums distinct must have its counting function up to N infinitely often much smaller than the cube root of N.
Asks whether every Sidon set in the first N integers extends to a Sidon set in a longer interval that is nearly as large as the largest possible.
Asks whether the density of integers whose totient is below a given fraction of the integer, as a function of that fraction, ever has a positive derivative.
Asks whether some infinite set of totient values has the smallest integer attaining each value growing faster than any fixed multiple of the value.
Asks whether the larger of the sumset and product set of a finite set of integers always has size at least the set's size squared, up to a small power loss.