1 research outputs found
Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers
For all nonnegative integers n, the Franel numbers are defined as 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.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func