trureturing — Discovering truth

bibkey: wu2026a221018
authors: Chai Wah Wu; Alex Ratushnyak
year: 2026
title: OEIS A221018, first FORMULA conjecture on adjacent-index Fibonacci divisibility
doi: null
url: https://oeis.org/A221018/internal
claim: "a(n) % 10 == 0, 2, or 8 (conjectured). - _Chai Wah Wu_, Aug 06 2026"
strata_touched:
  - D5/S3/Arith/Congruence/WuAdjacentFibonacciResidues
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A221018/internal

OEIS A221018, revision 35, Sep 09 2026 11:46:10, first FORMULA line. This web entry has no page number or recorded DOI. The conjecture is attributed to Chai Wah Wu; Alex Ratushnyak authored the sequence entry.

Source statement

The NAME line reads verbatim:

Number k such that Fibonacci(k) is divisible by k, k + 1 and k - 1.

The first FORMULA line reads verbatim:

a(n) % 10 == 0, 2, or 8 (conjectured). - Chai Wah Wu, Aug 06 2026

The generator begins def A221018_gen(startvalue=2): # generator of terms >= startvalue and ranges over integers at least two. Fibonacci indexing uses the standard sequence with F(0)=0 and F(1)=1. The Lean statement quantifies over natural numbers k with 2 ≤ k, retains each of the three divisibility hypotheses, and concludes k % 10 = 0 or k % 10 = 2 or k % 10 = 8. Subtraction k - 1 agrees with integer subtraction on this domain. The implication asserts a necessary condition and no converse.

Mathematical scope

The proof uses Fibonacci index divisibility at indices 3, 4 and 5. Divisibility by k and k+1 forces an even Fibonacci value; divisibility feedback then forces 12 to divide k. Divisibility by the neighbours excludes the remainders 1 and 4 modulo 5. The three stated decimal remainders follow.

Infinitude, density and infinitely many occurrences of each allowed final digit remain open. No downstream result relying on this conjecture was identified in the sources screened in issue #21590.

Finite evidence and literature boundary

The exact-arithmetic experiment is in the-omega-institute/trureturing-experiments, commit 69cc788c26a9c571f2142e468269a27521199775, at docs/reports/wu-a221018-fibdiv/check.py. Its SHA-256 is 3c291c0c963c2b0f20cd3320587422a4bde0879a032107daf2dfbb7917130c80. The reported command python3 docs/reports/wu-a221018-fibdiv/check.py exited 0 with ALL_CHECKS_OK. These finite checks support examples and source fidelity; they do not establish the universal implication.

Literature completeness is ASSUMED-UNVERIFIED. The screen in https://github.com/the-omega-institute/trureturing/issues/21590 found no settling reference in its searched scope. OpenAlex and Semantic Scholar citing lists were unavailable; Jarden was partly read and the Cihan–Inam journal version was not read. Negative identifier searches do not exclude an earlier proof published without the identifier.