Congruences for Franel numbers

Details Congruences for Franel numbers Congruences for Franel numbers

2011-12-05

arxiv.org/abs/1112.1034v10

Mathematics

Number Theory

Scientific paper

The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ (n=0,1,2,...) play important roles in both combinatorics and number theory. In the first paper "Congruences for Franel numbers", we initiate the investigation of fundamental congruences for Franel numbers, and mainly show the following congruences for any prime p>3: $$\sum_{k=0}^{p-1}(-1)^k*f_k=(p/3) (mod p^2), $$ $$\sum_{k=0}^{p-1}(-1)^k*kf_k=-2/3*(p/3) (mod p^2),$$ $$\sum_{k=1}^{p-1}(-1)^k*f_k/k =0 (mod p^2),$$ $$\sum_{k=1}^{p-1}(-1)^k*f_k/k^2 =0 (mod p).$$ The paper also contains several conjectural congruences.

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