Connections between p=x2+3y2 and Franel numbers

Abstract The Franel numbers are given by f n = ∑ k = 0 n ( n k ) 3 ( n = 0 , 1 , 2 , … ) . Let p > 3 be a prime. When p ≡ 1 ( mod 3 ) and p = x 2 + 3 y 2 with x , y ∈ Z and x ≡ 1 ( mod 3 ) , we show that ∑ k = 0 p − 1 f k 2 k ≡ ∑ k = 0 p − 1 f k ( − 4 ) k ≡ 2 x − p 2 x ( mod p 2 ) . We also prove that if p ≡ 2 ( mod 3 ) then ∑ k = 0 p − 1 f k 2 k ≡ − 2 ∑ k = 0 p − 1 f k ( − 4 ) k ≡ 3 p ( ( p + 1 ) / 2 ( p + 1 ) / 6 ) ( mod p 2 ) . In addition, we propose several related conjectures for further research.