Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

Details Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

2012-01-03

arxiv.org/abs/1201.0617v1

Journal of Number Theory 132 (2012) 1731-1740

Mathematics

Number Theory

8 pages

Scientific paper

10.1016/j.jnt.2012.02.004

For all nonnegative integers n, the Franel numbers are defined as $$
f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on
congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0
\pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2
\pmod{p^5}, where n is a positive integer and p>3 is a prime.

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