Erdos #488 density doublling. Fun elementary writeup + Wolfram Cloud verification (0 failures)

Ideas
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Found a little time to work on this weekend Project for Erdos #488 using wolfram software, grok 4.5 and my casio calculator. Erdős problem #488 (density doubling for sets of multiples) — still open on 488 | Erdős Problems with 0 claimed proofs.

Cloud census (SeedRandom[488]): 80 random primitive sets, window 120, totalFailures = 0.

Elementary writeup prepared for arXiv math.NT; looking for critical reads and anyone authorized to discuss endorsement for that category.

Problem: 488 | Erdős Problems

Weekendprojecterdos488Wolframan_Grok4.5 _OP_3_CP_.nb (6.27 MB)

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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):

F_A(m)/m < 2 F_A(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:

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.

Erdos488_Community_Companion.nb (12.5 KB)

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Just heard about Stephen Wolfram’s wife passing away. He has an amazing writeup of what she meant to him online. Sorry to hear that news. Inspiring woman it sounds like.