Proof of three conjectures on congruences

Abstract

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. It is shown that if p ≡ 1 (mod 4) or a > 1 then

$\sum\limits_{k = 0}^{\left\lfloor {\tfrac{3} {4}p^a } \right\rfloor } {\left( {\tfrac{{ - 1/2}} {k}} \right) \equiv \left( {\tfrac{2} {{p^a }}} \right)(\bmod p^2 )} , $

where (−) denotes the Jacobi symbol. This confirms a conjecture of the second author. A conjecture of Tauraso is also confirmed by showing that

$\sum\limits_{k = 1}^{\left\lfloor {p - 1} \right\rfloor } {\tfrac{{L_k }} {{k^2 }} \equiv 0(\bmod p)providedp > 5,} $

where the Lucas numbers L ₀, L ₁, L ₂, … are defined by L ₀ = 2, L ₁ = 1 and L _n+1 = L _n +L _n−1 (n = 1, 2, 3, …). The third theorem states that if p ≠ 5 then $F_{p^a - (\tfrac{{p^a }} {5})}$ mod p ³ can be determined in the following way:

$\sum\limits_{k = 0}^{p^a - 1} {( - 1)^k (_k^{2k} ) \equiv \left( {\tfrac{{p^a }} {5}} \right)(1 - 2F_{p^a - (\tfrac{{p^a }} {5})} )(\bmod p^3 ),} $

which appeared as a conjecture in a paper of Sun and Tauraso in 2010.