Towers of congruences for linear recurrences at unramified primes, with a proof of P. Bala's conjecture on the tribonacci numbers
Year: 2026
Type: preprint
Abstract: Let a(n) be a sequence of p-integral rationals satisfying a linear recurrence with monic characteristic polynomial f of degree d with integer coefficients, and let p be a prime not dividing the discriminant of f. Let D be the least common multiple of the degrees of the irreducible factors of f modulo p (the order of the Frobenius automorphism of the splitting field of f over Q_p). We prove that a(n p^(k-1+D)) ≡ a(n p^(k-1)) (mod p^(k + v_p(n))) for all n, k ≥ 1 and, when f is irreducible modulo p (D = d), the finer congruence Sum_{j=0}^{d-1} a(n p^(k-1+j)) ≡ a(0) · s(n p^(k-1)) (mod p^(k + v_p(n))), where s(m) is the sum of the m-th powers of the roots of f; for f reducible modulo p the sum over a full Frobenius period decomposes along the p-adic factorization of f. The case d = 3, f = x^3 − x^2 − x − 1, a(0) = 0 proves the conjecture stated by P. Bala in the OEIS entry A000073 of the tribonacci numbers (December 2022), that T(n p^(k+1)) + T(n p^k) + T(n p^(k-1)) ≡ 0 (mod p^k) for the primes p of A106282, and improves its modulus to p^(k + v_p(n)); for the other primes p ≠ 2, 11 it gives T(n p^(k+1)) ≡ T(n p^(k-1)) or T(n p^k) ≡ T(n p^(k-1)) (mod p^(k + v_p(n))) according as x^3 − x^2 − x − 1 has one or three roots modulo p. For the Fibonacci and Pell numbers the inert case reads U(n p^k) ≡ −U(n p^(k-1)) (mod p^(k + v_p(n))) (p ≡ ±2 mod 5, resp. p ≡ ±3 mod 8), and the split case U(n p^k) ≡ U(n p^(k-1)); for the Padovan numbers the sum of three terms is congruent to a Perrin number. The proofs are short applications of the Frobenius automorphism of the unramified extensions of Q_p. The exponent k + v_p(n) is attained for every prime tested except at the Pell–Wieferich primes 13 and 31, and the ramified primes are shown to be genuine exceptions (tribonacci at p = 2, 11; Fibonacci at p = 5; Pell at p = 2). The conjecture was selected by a scan of a local copy of the OEIS for open "tower" conjectures, the type that the author's SyntheticMind engine attacks and generalizes; the engine's exponent-form fitting proposed the refinement k + v_p(n) before the proof was written, and the method (fitting the recurrence, factoring f modulo p, classifying the primes, proving the inert case) has been implemented in the engine. The proofs and the text were prepared with the assistance of an AI system under the author's direction; the author checked every step and takes full responsibility for the content. Files: Blanco_Gomez_2026_recurrence_towers_unramified_EN.pdf is the article (6 pages); Blanco_Gomez_2026_torres_recurrencias_no_ramificados_ES.pdf is the Spanish version (same content); verify_recurrence_towers.py is an independent verification script (Python 3, exact integers, standard library only) that computes seven sequences and their power sums from scratch, factors f modulo every prime p ≤ 59, checks the three theorems with their exact exponent (the general one by Hensel lifting of the factorization and traces in Z_p[x]/(F_l)) and runs the negative controls; verify_recurrence_towers.log is its output (ALL CHECKS PASSED).
Author: Blanco, Gómez
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222483259
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