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Proof of three conjectures on congruences
- Source :
- Science China Mathematics. 57:2091-2102
- Publication Year :
- 2014
- Publisher :
- Springer Science and Business Media LLC, 2014.
-
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 $p\equiv 1\pmod{4}$ or $a>1$ then $$ \sum_{k=0}^{\lfloor\frac34p^a\rfloor}\binom{-1/2}k\equiv\left(\frac{2}{p^a}\right)\pmod{p^2}, $$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\sum_{k=1}^{p-1}\frac{L_k}{k^2}\equiv0\pmod{p}\quad {\rm provided}\ \ p>5,$$ where the Lucas numbers $L_0,L_1,L_2,\ldots$ are defined by $L_0=2,\ L_1=1$ and $L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots)$. Our third theorem states that if $p\not=5$ then we can determine $F_{p^a-(\frac{p^a}5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{p^a-1}(-1)^k\binom{2k}k\equiv\left(\frac{p^a}5\right)\left(1-2F_{p^a-(\frac{p^a}5)}\right)\ \pmod{p^3},$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.<br />16 pages, final published version
- Subjects :
- Discrete mathematics
Conjecture
Fibonacci number
Mathematics - Number Theory
11B65, 11A07, 05A10, 11B39
Lucas sequence
General Mathematics
Congruence relation
Integer
Lucas number
FOS: Mathematics
Mathematics - Combinatorics
Number Theory (math.NT)
Combinatorics (math.CO)
Jacobi symbol
Binomial coefficient
Mathematics
Subjects
Details
- ISSN :
- 18691862 and 16747283
- Volume :
- 57
- Database :
- OpenAIRE
- Journal :
- Science China Mathematics
- Accession number :
- edsair.doi.dedup.....d97c66912b02a76654041102b3fd29f4