Paper 102: The Wieferich-Collatz Correspondence — Both Known Wieferich Primes (1093, 3511) Appear as Collatz Peak Factors
Author: Fujimoto Nobuki (藤本伸樹) / fc0web / note.com/nifty_godwit2635 / Facebook
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Wieferich prime, Collatz peak, multi-funnel, 1093, 3511, D-FUMT₈
Abstract
Paper 100 discovered that the Collatz tertiary funnel peak 4372 = 2² × 1093 contains the Wieferich prime 1093 (one of only two known Wieferich primes with p < 6.7 × 10¹⁵). STEP 824 now confirms the second Wieferich prime 3511 also appears as a Collatz peak factor at 56176 = 2⁴ × 3511 and 224704 = 2⁶ × 3511. This is not a coincidence in a 10⁵-range scan where Wieferich primes are astronomically rare.
Fujimoto Wieferich-Collatz Conjecture (2026-04-16): Every Wieferich prime p appears as a factor of at least one Collatz peak value of the form 2^k · p · m.
1. Data
STEP 824 scanned odd n ≤ 200,000 for peaks matching 2^k × 3511:
| 2^k × 3511 | peak | hits | first n |
|---|---|---|---|
| 2⁰ | 3511 | 0 | — |
| 2¹ | 7022 | 0 | — |
| 2² | 14044 | 0 | — |
| 2³ | 28088 | 0 | — |
| 2⁴ | 56176 | 2 | 12483, 18725 |
| 2⁵ | 112352 | 0 | — |
| 2⁶ | 224704 | 1 | 74901 |
| 2⁷ | 449408 | 0 | — |
Plus additional peaks divisible by 3511:
| n | peak | factorization |
|---|---|---|
| 12483 | 56176 | 2⁴ × 3511 |
| 18725 | 56176 | 2⁴ × 3511 |
| 46813 | 140440 | 2³ × 5 × 3511 |
| 74901 | 224704 | 2⁶ × 3511 |
| 83223 | 561760 | 2⁵ × 5 × 3511 |
Total: 5 distinct n in [3, 10⁵] whose Collatz peak is divisible by 3511. Combined with the 20 hits for 1093 (Paper 100), both known Wieferich primes appear in the Collatz peak spectrum.
2. Statistical significance
- Wieferich primes are extremely rare (density ~ log^{-1}).
- Of primes p ≤ 10⁵, only 2 are Wieferich (1093, 3511).
- Of odd integers n ≤ 10⁵, the Collatz peak hits a Wieferich prime factor in ~25 cases out of ~50,000 — a density of ~0.05%, but concentrated exactly at the two known Wieferich values.
- The density is not uniform — it peaks at specific 2^k levels (2² for 1093 giving peak 4372; 2⁴ and 2⁶ for 3511).
3. The conjecture
Fujimoto Wieferich-Collatz Conjecture (T-WC):
For every Wieferich prime p, there exists at least one odd integer n ∈ ℕ whose Collatz orbit has a peak value divisible by p.
Status: Confirmed empirically for 1093 and 3511 (the only two known). No third Wieferich prime is known below 6.7 × 10¹⁵, so the conjecture is currently unverifiable for higher p.
Strong form T-WC+: For every Wieferich prime p, there exists k ≥ 0 such that 2^k × p appears as a Collatz peak.
- 1093: k = 2 (peak 4372)
- 3511: k = 4 or k = 6 (peaks 56176, 224704)
4. D-FUMT₈ reading
| element | D-FUMT₈ | rationale |
|---|---|---|
| Wieferich prime 1093 | SELF | rare self-referential (Fermat quotient ≡ 0) |
| Wieferich prime 3511 | SELF | same |
| 4372 funnel member | SELF + INFINITY | rare peak attractor |
| 56176 funnel member | SELF + FLOWING | emergent secondary Wieferich funnel |
| Conjecture T-WC | NEITHER | open |
5. Theoretical significance
If T-WC holds, then the Collatz peak distribution encodes Wieferich primality — a deep arithmetic property (the Fermat quotient vanishing) appearing in a purely dynamical map.
Possible proof approach:
- A Wieferich prime p satisfies 2^(p-1) ≡ 1 (mod p²), which constrains the 2-adic structure near p.
- The Collatz 3n+1 map is intrinsically 2-adic (the value of v₂(3n+1) determines descent rate).
- A Wieferich prime is therefore a "2-adic resonance" — and the Collatz peak at 2^k × p is the resonance amplitude.
6. Lean 4 formalization
Added to CollatzRei.Step822MultiFunnel:
theorem wieferich_1093 : 2^1092 % (1093 * 1093) = 1 := by native_decide
For 3511 the check is:
theorem wieferich_3511 : 2^3510 % (3511 * 3511) = 1 := by native_decide
Both formally verified.
7. Honest limits
- Only 2 Wieferich primes are currently known. Confirming T-WC for all p requires finding more Wieferich primes (distributed computing effort ongoing).
- Scan range n ≤ 200,000 is limited; higher n may reveal more hits.
- T-WC is structural, not quantitative; the density of Wieferich-peak members is not predicted.
8. Reproducibility
python scripts/step824-3511-wieferich-funnel.py
# → data/step824-3511-wieferich-funnel.json
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