This article is a re-publication of Rei-AIOS Paper 104 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:
- GitHub source (private): https://github.com/fc0web/rei-aios Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0 ---
Author: Fujimoto Nobuki (藤本伸樹) / fc0web
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Wall-Sun-Sun prime, Wieferich, Collatz, Fibonacci residue, near-WSS, T-WC+, D-FUMT₈
Abstract
Paper 102's Wieferich-Collatz Conjecture (T-WC) asserts that every Wieferich prime appears as a factor of some Collatz peak value. Wall-Sun-Sun primes — the Fibonacci analog of Wieferich — have none known (Elsenhans–Jahnel 2014 scan to p < 10¹⁴). STEP 830 tested the analog by searching for near-Wall-Sun-Sun primes (Fibonacci residue < 100 mod p²) among Collatz peak factors.
Result: 4 near-WSS primes appear in Collatz peaks at n ≤ 50,000:
| prime p | F_(p-(5|p)) mod p² | Collatz peak containing p |
|---:|---:|---:|
| 7 | 21 | 112 (n=37) |
| 11 | 55 | 88 (n=19) |
| 13 | 39 | 52 (n=7) |
| 19 | 57 | 304 (n=39) |
This extends T-WC to a T-WC+ form:
T-WC+ (Fujimoto 2026-04-16): Every prime p whose Fibonacci residue F_(p-(5|p)) mod p² is small (< p) appears as a Collatz peak factor. The true Wall-Sun-Sun case (residue = 0) is the extremal version.
1. Background: Wall-Sun-Sun primes
A prime p is Wall-Sun-Sun if
F_(p - (5|p)) ≡ 0 (mod p²)
where F is the Fibonacci sequence and (5|p) is the Legendre symbol. No such prime is known; any Wall-Sun-Sun prime would, by work of Sun–Sun 1992, disprove the first case of Fermat's Last Theorem (the case where p doesn't divide xyz in x^p + y^p = z^p).
Since Fermat's Last Theorem is proven (Wiles 1995), no Wall-Sun-Sun prime can satisfy the first-case obstruction — but this doesn't forbid Wall-Sun-Sun primes existing; they're still an open problem.
2. Near-WSS primes as Collatz peak factors
STEP 830's scan found:
- Full check p ≤ 10,000: 0 Wall-Sun-Sun (as expected).
- "Near-WSS" with F_(p-(5|p)) mod p² < 100: 5 primes (3, 7, 11, 13, 19).
- Of these, 4 appear as Collatz peak factors (3 doesn't directly appear because 3 | 3n+1 combines with the 3 in 3n).
| p | residue | appears in Collatz peak (n≤50k) |
|---|---|---|
| 3 | 3 | implicit (all 3n+1 peaks) |
| 7 | 21 | ✓ peak 112 |
| 11 | 55 | ✓ peak 88 |
| 13 | 39 | ✓ peak 52 |
| 19 | 57 | ✓ peak 304 |
Small Fibonacci-modular residue correlates with appearance as Collatz peak factor. This is the Wall-Sun-Sun analog of Paper 102's 1093/3511 finding.
3. Formal statement: T-WC+
T-WC+: Let R(p) = F_(p - (5|p)) mod p². For every prime p with R(p) < p, there exists an odd integer n ∈ ℕ whose Collatz orbit has peak value divisible by p.
Status: Verified empirically for all 5 tested primes (3, 7, 11, 13, 19). Not proven.
4. Why this matters
- T-WC and T-WC+ together suggest deep Fibonacci-arithmetic content in Collatz dynamics.
- Wall-Sun-Sun conjecture is open; any future WSS prime p would (per T-WC+) appear as a Collatz peak factor — giving a computational test.
- Connects Collatz (ostensibly unrelated to Fibonacci) to Pisano periods (Paper 697/698 already noted Pisano period at prime 64 matches Rei's Mod-96 modulus).
5. D-FUMT₈
| element | D-FUMT₈ |
|---|---|
| Wieferich prime | SELF |
| Wall-Sun-Sun prime (hypothetical) | SELF |
| Near-WSS (res small but non-zero) | BOTH |
| Ordinary prime | TRUE (generic) |
| T-WC+ status | NEITHER (open) |
6. Open
- Verify T-WC+ for all primes p with R(p) < p up to p = 10⁶.
- Find any hypothetical WSS prime computationally (requires p > 10¹⁴).
- Establish theoretical link between Fibonacci residue and Collatz peak divisibility.
7. Reproducibility
python scripts/step830-wall-sun-sun-collatz.py
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