On finite differences, permutations, and Quadratic residues

AbstractLet p be an odd prime and n an integer relatively prime to p. In this work three criteria which give the value of the Legendre symbol (np) are developed. The first uses two adjacent rows of Pascal's triangle which depend only on p to express (np) explicitly in terms of the numerically least residues (mod p) of the numbers n, 2n, …, [(p + 1)4]n or of the numbers [(p + 1)4]n,…, [(p − 1)2]n. The second, analogous to a theorem of Zolotareff and valid only if p ≡ 1 (mod 4), expresses (np) in terms of the parity of the permutation of the set {1,2,…, ((p− 1)2} defined by the absolute values of the numerically least residues of n, 2n,…,[(p− 12]n. The third is a result dual to Gauss' lemma which can be derived directly without Euler's criterion. The applications of the dual include a proof of Gauss' lemma free of Euler's criterion and a proof of the Quadratic Reciprocity Law