A proof of the Auluck–Chowla–Gupta Conjecture

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), …, p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n ≥ 1. For n ≥ 10⁵ we give an analytic argument with fully explicit constants, certified in ball arithmetic; for n ≤ 2·10⁵ the conjecture is verified exhaustively by two independent programs, one using exact integer arithmetic only.