Michael Schroeder · Preprint · Version 1.0.1 · 1 September 2026 · ORCID 0009-0004-3249-0195

Seven Prime Divisors in Odd Distinct Covering Systems

Abstract

More than seventy-five years after Erdős introduced distinct covering systems in 1950, the Erdős–Selfridge odd covering problem (Erdős Problem 7) remains open: does there exist a finite covering system with pairwise distinct odd moduli greater than one? We prove that the least common multiple N of the moduli in any such system has at least seven distinct prime divisors, without any squarefreeness or prime-power-height hypothesis. The proof passes to a residual product of prime trees on which all prime-power classes are absent, then charges every remaining cylinder at the last digit that it fixes. A normalized fibre distortion and a reverse causal comparison for increasing supermodular functionals reduce the estimate to five exact rational charges. Their sum is 0.914634…<1, giving the contradiction uniformly in all prime-power heights. A complete Lean 4 formalization and an independent exact rational checker accompany the argument.

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Version 1.0.1 was revised on 14 September 2026 to add publication identifiers and clarify reproduction instructions. The mathematical content and Lean/Python verification sources are unchanged from version 1.0.0.

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Explanation of the problem

A congruence class is a regularly spaced set of integers. For example, the class 1 modulo 3 is …, −5, −2, 1, 4, 7, …: all integers with remainder 1 on division by 3. A covering system is a finite collection of such classes whose union contains every integer.

Each spacing is called a modulus. The Erdős–Selfridge odd covering problem asks whether a covering is possible when every modulus is odd and greater than one, and no two classes have the same modulus. Classes may overlap; every integer must be covered at least once.

The rank is the number of different primes appearing across all moduli, equivalently the number of distinct prime divisors of their least common multiple. The set of these primes is called the prime support. For example, moduli 9 and 25 use only the primes 3 and 5; raising them to higher powers does not add new primes.

This paper proves that any odd distinct covering system must use at least seven distinct primes. Equivalently, every finite candidate supported on six or fewer primes misses some integers, however large the prime powers are. This gives a necessary condition for a covering; the general existence question remains open.