Supercongruences involving Lucas sequences

2016/10/11 by Zhi‐Wei Sun, Sun, Zhi-Wei
Mathematics · Physics and Astronomy · #05A10 #11A07 #11B39 #11B65 #11B75 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1610.03384

openalex publication_date 2016/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For A,B∈\mathbb Z, the Lucas sequence u_n(A,B) (n=0,1,2,…) are defined by u₀(A,B)=0, u₁(A,B)=1, and u_n+1(A,B) = Au_n(A,B)-Bu_n-1(A,B) (n=1,2,3,…). For any odd prime p and positive integer n, we establish the new result \fracu_pn(A,B) - (\fracA²-4Bp) u_n(A,B)pn ∈ \mathbb Z_p, where (\frac⋅p) is the Legendre symbol and \mathbb Z_p is the ring of p-adic integers. Let p be an odd prime and let n be a positive integer. For any integer m\not≡0\pmod p, we show that \frac1pn(∑_k=0^pn-1 \frac\binom2kkm^k -(\fracΔp) ∑_r=0^n-1\frac\binom2rrm^r)∈\mathbb Z_p and furthermore \frac1n(∑_k=0^pn-1 \frac\binom2kkm^k -(\fracΔp) ∑_r=0^n-1\frac\binom2rrm^r)≡ \frac\binom2n-1n-1m^n-1 u_p-(\fracΔp)(m-2,1) \pmodp² where Δ=m(m-4). We also pose some conjectures for further research.

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