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Article overview
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Proof of three conjectures on congruences | | Hao Pan
; Zhi-Wei Sun
; | | Date: |
13 Oct 2010 | | Abstract: | 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 $$ sum_{k=0}^{[3p^a/4]}inom{-1/2}{k}=(2/p) (mod p^2),$$ 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 $p
ot=5$ then we can
determine $F_{p^a-(p^a/5)}$ mod $p^3$ in the following way:
$$sum_{k=0}^{p^a-1}(-1)^kinom{2k}{k}=(p^a/5)(1-2F_{p^a-(p^a/5)}) (mod
p^3),$$ which appeared as a conjecture in a paper of Sun and Tauraso [Adv. in
Appl. Math. 45(2010)]. | | Source: | arXiv, 1010.2489 | | Services: | Forum | Review | PDF | Favorites | |
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