Open Research Index

Erdős 689: eventual double covering by prime residue classes

Linmiao Xu

Palomar Registrypalomar/PALOMAR-2026-09-20-000002Sep 20, 2026math.NTscore 0

A complete Lean 4 proof of Erdős problem 689: for all sufficiently large n, one residue class per prime p ≤ n can cover every integer from 1 through n at least twice. The proof follows the argument for Theorem 1.1 in Chojecki’s April 2026 manuscript “A greedy matching proof of Erdős’s two-fold residue-class problem.” The formalization directly proves the required three-prime counting estimate using Fourier analysis, replacing the manuscript’s invocation of Green–Tao, and verifies the sieve bounds and greedy-matching construction in Lean.

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