Quadratic residues and related permutations and identities

Abstract

Let p be an odd prime. In this paper we investigate quadratic residues modulo p and related permutations, congruences and identities. If a1<…<a(p−1)/2 are all the quadratic residues modulo p among 1,…,p−1, then the list {12}p,…,{((p−1)/2)2}p (with {k}p the least nonnegative residue of k modulo p) is a permutation of a1,…,a(p−1)/2, and we show that the sign of this permutation is 1 or (−1)(h(−p)+1)/2 according as p≡3(mod8) or p≡7(mod8), where h(−p) is the class number of the imaginary quadratic field Q(−p). To achieve this, we evaluate the product ∏1⩽j<k⩽(p−1)/2(cot⁡πj2/p−cot⁡πk2/p) via Dirichlet's class number formula and Galois theory. We also obtain some new congruences and identities in product forms; for example, we determine the exact value of∏1⩽j<k⩽p−1cos⁡πaj2+bjk+ck2p for any a,b,c∈Z with ac(a+b+c)≢0(modp).