Supercongruences on some binomial sums involving Lucas sequences

TL;DR

This paper proves several conjectured supercongruences involving binomial sums and Lucas sequences, using hypergeometric transformations, and introduces new related congruences.

Contribution

It confirms several conjectured supercongruences and proposes new congruences involving binomial sums and Lucas sequences, expanding the understanding of divisibility properties.

Findings

Proved that certain binomial sums involving Pell numbers are divisible by p^2 for primes p ≡ 7 mod 8.

Confirmed conjectures posed by Sun on binomial sum divisibility.

Proposed three new supercongruences related to Lucas sequences.

Abstract

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.