MS / SG

Michael Schroeder · Preprint · Version 6 · 6 September 2026

Eight Prime Divisors in Odd Distinct Covering Systems

Every finite covering of the integers by congruence classes with pairwise distinct odd moduli greater than one has a least common multiple divisible by at least eight distinct primes. Prime-power exponents are unrestricted.

The paper also establishes a ceiling for its specified scalar method. This limits that method, not the possible rank of a covering system; the Erdős–Selfridge odd covering problem remains open.

Paper and guide

The guide states the two formal theorems, their trust boundaries and the reproduction steps. It distinguishes the fresh covering-proof replay from the preserved large scalar replay. This is a preprint, not a claim of peer review or second-platform verification.

Sources and complete companion

Download the compact source ZIP (777 KiB): LaTeX, both Lean projects, exact inputs, deterministic certificate generators and supporting documentation.

The complete companion additionally includes the expanded certificates and original verification logs. It is the unchanged publication ZIP, 100.9 MiB. To fit the website’s per-file limit, it is stored in five parts; the button checks each part and the complete SHA-256 digest before offering the original ZIP.

The full download needs approximately 101 MiB of network data and several hundred MiB of available browser memory.

Command-line download and manual reconstruction

Download download_companion.py and run python3 download_companion.py. It uses only Python’s standard library, verifies each part and the finished archive, and will not overwrite an existing output file.

Alternatively, save all five parts in the same folder:

  1. Part 1 · 24,000,000 bytes
  2. Part 2 · 24,000,000 bytes
  3. Part 3 · 24,000,000 bytes
  4. Part 4 · 24,000,000 bytes
  5. Part 5 · 9,789,211 bytes

On macOS or Linux, concatenate the parts in this exact order:

cat eight_prime_support_companion_v6.zip.part01 \
    eight_prime_support_companion_v6.zip.part02 \
    eight_prime_support_companion_v6.zip.part03 \
    eight_prime_support_companion_v6.zip.part04 \
    eight_prime_support_companion_v6.zip.part05 \
    > eight_prime_support_companion_v6.zip
shasum -a 256 eight_prime_support_companion_v6.zip

The result must have SHA-256:

f21fb6f72dbba419adf151243adc02cd91389de430821957309594da7a98b858

Hashes identify the released bytes, not mathematical correctness. Check the extracted archive’s own SHA256SUMS before following its build instructions.

Citation, context and provenance

BibTeX · Citation metadata · Release metadata · Checksums · Rights status

This work strengthens the author’s rank-seven theorem, credited as antecedent work but not assumed in the rank-eight proof. No arXiv identifier or DOI has been assigned in this release.

Keywords: Erdős–Selfridge odd covering problem; Erdős Problem 7; covering systems; prime divisors; supermodularity; distortion method; formal verification.

Acknowledgments

This work benefited from research assistance by AI systems developed by OpenAI and Anthropic, including support for proof exploration, proof development, and exact computational checks among others.