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Ramanujan congruences for the partition function

Carter Hastings

Palomar Registrypalomar/PALOMAR-2026-10-22-000007Oct 22, 2026math.NTscore 0

A Lean formalization of modular forms over coefficient subrings of the complex numbers, their coefficientwise reductions modulo ring homomorphisms, and a proof classifying the Ramanujan congruences for the partition function following an argument from Ahlgren and Boylan. The recorded results prove Ramanujan's congruences modulo 5, 7, and 11 as well as the fact there there are no other such examples. The proof develops modular forms modulo a prime, filtration, and theta and U operators, using the arguments of Ahlgren–Boylan, Kiming–Olsson, and Swinnerton-Dyer.

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