Proof of three conjectures on congruences

Details Proof of three conjectures on congruences Proof of three conjectures on congruences

2010-10-12

arxiv.org/abs/1010.2489v3

Mathematics

Number Theory

16 pages. Polished version

Scientific paper

In this paper we prove three conjectures on congruences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p=1 (mod 4)$ or $a>1$ then $$ \sum_{k=0}^{[3p^a/4]}\binom{-1/2}{k}=(2/p^a) (mod p^2),$$ where (-) denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraos by showing that $\sum_{k=1}^{p-1}L_k/k^2=0 (mod p^2)$ if $p>5$, where the Lucas numbers $L_0,L_1,L_2,...$ are defined by $L_0=2, L_1=1$ and $L_{n+1}=L_n+L_{n-1} (n=1,2,3,...)$. Our third theorem states that if $p\not=5$ then we can determine $F_{p^a-(p^a/5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{p^a-1}(-1)^k\binom{2k}{k}=(p^a/5)(1-2F_{p^a-(p^a/5)}) (mod p^3),$$ which appeared as a conjecture in a paper of Sun and Tauraso [Adv. in Appl. Math. 45(2010)].

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