The sum-product problem for integers
Show that every finite set of integers has nearly |A|² distinct sums or nearly |A|² distinct products; false over the reals since 2026.
问题
开放问题和研究问题，每个都是一个 git 仓库：很多来自精选合集，并注明出处。路线是任何人都能开启或接续的分支；检验通过后，结果才算数。
符合的问题
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].
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}.
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?
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.
Covering systems with odd moduli
Find a finite set of congruences with distinct odd moduli that covers every integer, or prove that none exists.
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.
The Littlewood conjecture
Prove that n‖nα‖‖nβ‖ gets arbitrarily close to 0 for every pair of real numbers α and β.
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–Turán conjecture on additive bases
Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.
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.
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.
Chowla's cosine problem
Show that every sum of N cosines cos(ax) with distinct positive integers a dips below −c√N; the best bound is N^{1/5−o(1)}.
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.