Connections between p = x2+3y2 and Franel numbers

The Franel numbers are given by f(n) = Sigma(n)(k=0) ((n)(k))(3) (n = 0, 1, 2, ... ). Let p > 3 be a prime. When p equivalent to 1 (mod 3) and p = x(2) + 3y(2) with x, y is an element of Z and x equivalent to 1 (mod 3), we show that Sigma(p-1)(k=0) f(k)/2(k) equivalent to Sigma(p-1)(k=0) f(k)/(-4)(k) equivalent to 2x - p/2x (mod p(2)). We also prove that if p equivalent to 2 (mod 3) then Sigma(p-1)(k=0) f(k)/2(k) equivalent to -2 Sigma(p-1)(k=0) f(k)/(-4)(k) equivalent to 3p/((p+1)/2(p+1)/6) (mod p(2)). In addition, we propose several related conjectures for further research. (c) 2013 Elsevier Inc. All rights reserved.