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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