Proof of three conjectures on congruences

Abstract

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1(\mathrm{mod}4)$ or $a>1$ then $$\sum _{k=0}^{\lfloor \frac{3}{4}{p}^a\rfloor }\left(\genfrac{}{}{0ex}{}{-1/2}{k}\right)\equiv \left(\frac{2}{p^a}\right)(\mathrm{mod}{p}^2),$$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\sum _{k=1}^{p-1}\frac{L_k}{k^2}\equiv 0(\mathrm{mod}p)\mathrm{provided}p>5,$$ where the Lucas numbers $L_0,L_1,L_2,\dots$ are defined by $L_0=2,L_1=1$ and $L_{n+1}=L_n+L_{n-1}(n=1,2,3,\dots )$. Our third theorem states that if $p\ne 5$ then we can determine $F_{p^a-(\frac{p^a}{5})}$ mod $p^3$ in the following way: $$\sum _{k=0}^{p^a-1}(-1{)}^k(\genfrac{}{}{0ex}{}{2k}{k})\equiv \left(\frac{p^a}{5}\right)\left(1-2F_{p^a-(\frac{p^a}{5})}\right)(\mathrm{mod}{p}^3),$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.

16 pages, final published version

Cited by in corpus (2)

• Open Conjectures on Congruences
• Fibonacci numbers modulo cubes of primes