bibkey: bala2022a144837 authors: Peter Bala year: 2022 title: “OEIS A144837, Lucas(5^n), with the 5-adic congruence conjecture a(n+1) ≡ a(n) (mod 5^(n+r+1))” doi: null url: https://oeis.org/A144837 claim: “%N a(n) = Lucas(5^n). %F Conjecture: a(n+1) == a(n) (mod 5^(n+r+1)) for n >= r.” strata_touched:

• D5/S1/Recurrence/BalaLucasFiveAdicCongruence license: citation-only triage: anchor

OEIS A144837

The associated theorem proves Peter Bala’s conjectured congruence for every natural n and every natural r <= n, using his integer recurrence. The extension a(0)=1=Lucas(5^0)=Lucas(1) gives a(1)=11=A144837(1) and agrees with the OEIS domain n >= 1. Congruence is expressed as divisibility of the integer difference by 5^(n+r+1). Only the Conjecture line is settled; the Lucas representation, the 5-adic limit (A269591), and A268922 are not claimed. Artur Jasinski authored the entry; Bala authored the conjecture in the Nov 14 2022 block.

Verified locator

• URL: https://oeis.org/A144837
• %N (verbatim): a(n) = Lucas(5^n).
• %O (verbatim): 1,1
• %A (verbatim): _Artur Jasinski_, Sep 22 2008
• Peter Bala’s Nov 14 2022 recurrence (verbatim): a(n) = a(n-1)^5 + 5*a(n-1)^3 + 5*a(n-1) with a(1) = 11.
• Context in the same block (verbatim): a(n+1) == a(n) (mod 5^(n+1)) for n >= 1 (a particular case of the Gauss congruences for the Lucas numbers).
• Settled %F line (verbatim): Conjecture: a(n+1) == a(n) (mod 5^(n+r+1)) for n >= r.
• Source transcription and literature readings: 2026-09-15; ASSUMED-UNVERIFIED without independent online retrieval.

According to these readings, no literature proof of the strengthening was found in the searched surfaces: Bala’s block proves only mod 5^(n+1), and his 2026 note on strong divisibility sequences concerns a(n+k) - a(k).