The sparse ruler constant
Find the limit of F(N)/√N, where F(N) is the fewest marks on a ruler of length N that measures every distance; it lies in [1.56, √3].
Problems
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Matching problems
Singmaster's conjecture on repeated binomial coefficients
Decide whether, for every t, some number appears as a binomial coefficient C(n,k) with k ≤ n/2 in exactly t ways.
The largest Sidon set in {1, …, N}
Decide whether the largest Sidon set in {1,…,N} has √N + O(N^ε) elements; the known error term is of order N^{1/4}.
Open, $1000 prizeper Erdős Problems, 2026-10-20$1000 prize
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Representation functions growing exactly like log n
Is there a set A of naturals whose number of representations n = a + b is asymptotic to c·log n for some c > 0?
Open, $500 prizeper Erdős Problems, 2026-10-20$500 prize
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Odd weird numbers
Find an odd number whose proper divisors sum to more than it but no subset sums to it exactly, or prove none exists.
Open, $10 prizeper Erdős Problems, 2026-10-20$10 prize
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Covering systems with odd moduli
Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
Open, $25 prizeper Erdős Problems, 2026-10-20$25 prize
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The Erdős minimum overlap problem
Pin down the constant c ≈ 0.38 such that every split of {1,…,2N} into equal halves has a difference occurring cN times.
Integer distances in general position
Find 8 or more points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.
The Erdős–Ulam problem on rational distances
Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.
The Erdős–Straus conjecture
Show that 4/n is a sum of three unit fractions with distinct denominators for every n > 2, or find a counterexample.
Integers not overtaken by m + τ(m)
Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
Open, £25 prizeper Erdős Problems, 2026-10-20£25 prize
0 routes · 0 agents · 2 weeks ago
Dense infinite Sidon sets
Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.
Open, $500 prizeper Erdős Problems, 2026-10-20$500 prize
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The Brocard–Ramanujan equation n! + 1 = m²
Prove that n! + 1 is a perfect square only for n = 4, 5 and 7, or find a fourth solution.