That's an interesting project! I hope your write-up turns out well. It would be great to see the details behind your approach, especially how you tested it with the random primitive sets. Before submitting to arXiv, I'd recommend getting feedback from a few number theory researchers, as fresh eyes can often spot gaps or suggest improvements. Best of luck with the proof and the endorsement process!

Thanks — that's fair feedback, and I agree on the arXiv point.

You're right that "0 failures" alone doesn't say much. Here's what the Cloud notebook actually does, and how the writeup is structured.

The inequality (#488)
For finite nonempty A and all m > n >= max(A):

FA(m)/m < 2 FA(n)/n

F_A(x) counts integers <= x divisible by some a in A. Constant 2 is sharp (near A={5}, n=9).

How the random primitive-set check works
(reproducible: SeedRandom[488])

• Draw random sets from 2..maxA, keep only primitive ones (no element divides another).
• For each set A, set n0 = Max[A], then check m = n0+1 .. n0+window.
• Pass/fail is exact integer compare: F(m)n < 2F(n)*m (not floats).
• Default light run: 30 sets, maxA=20, kmax=4, window=60. I get totalFailures=0 in ~0.3s.
• Earlier post mentioned a heavier 80-set / window-120 run — same idea, bigger sample. Still not a proof.

The notebook prints every set in the sample, a min-slack histogram, sieve strips, and a near-sharp singleton plot. If anyone finds a failing (A,n,m), please post it.

Sketch of the writeup (not a TeX dump)
Primitive reduction, then a key lemma on 2F-E, reduce to an excess quantity Q, then induct on a smaller "EX2" statement. Hard residual goes through SD + L* (lattice). I killed a ratio bound R on a small counterexample (B={8,12,14}, m=15, c=20, q=9) — that bound is not used. The densest hand step for a reader is the Charge bookkeeping with free multi-points. A shorter paper could claim |A|<=6 first via the size ladder.

I treat the notebook as checks and pictures, not a referee. Zero census failures is evidence, not a theorem.

arXiv
Agreed: better to get a few number-theory reads before submitting. Happy to take heat on Charge, the induction order, or whether |A|<=6 should go first. WL speedups also welcome (the census is straightforward Floor/LCM inclusion-exclusion).

Also noting the Editorial Board note — the notebook is the WL side: exact F_A, plots, census. Source paste is in the Cloud notebook; I can attach the short .wl if useful.

Cloud notebook:
https://www.wolframcloud.com/env/a60/Weekendprojecterdos488Wolframan_Grok4.5.nb

Problem:
https://www.erdosproblems.com/488

Prior art I'm building from: Blair (pairs), MalekZ (2 in A / families), Chojecki-type small |A| work. Correct me if I miscredit.

Thanks again for the push to show the method.

Attachments:

Erdos488_Communi...nb