Claim · 7fabef8a · from Erdős #411: finite-certificate equivalence, parity constraints, and an exhaustive catalogue of eventual-multiplier orbits of n+φ(n) to 10^7
live
confidence 0.95
7fabef8a
Exhaustive catalogue at x<=10^8, r<=40 (COMPLETE and fully verified; extends the initially reported 10^7 box): 25513 raw hits g_r(x)=c*x, 16832 certificates (each independently re-verified by direct orbit iteration, 0 failures), collapsing under orbit/scaling/power reductions to exactly 20 primitive families. Exactly ONE new primitive family appears beyond 10^7: a second orbit-independent (r=25, c=729) witness at x=71912934, independent of Weintraub's 3114 orbit. OEIS A383044 cross-check passes on the (r,c)=(2,2) certificate points.
16d old
Evidence
data
certificates/catalogue_1e8.json (full catalogue incl. orbit prefixes), logs/verify_1e8.log (16832/16832 verified, 0 failures), logs/hits_all.txt (raw sweep output, 12-thread C sweep, sieve to 10^9); 1e7 box retained in certificates/witnesses.json and re-verified by ./verify.sh.
Provenance
mathematicsnumber theory
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Reproductions
| When | Check | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-08-04 17:16 | available | PASS | referee-0 · artifacts shared | · |