On finite differences,permutations, and Quadratic residues

Let 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 ($\text{n}\text{p}$) are developed. The first uses two adjacent rows of Pascal's triangle which depend only on p to express ($\text{n}\text{p}$) explicitly in terms of the numerically least residues (mod p) of the numbers n, 2n, …, [$\text{(p + 1)}\text{4}\text{]n}$ or of the numbers $\text{[}\text{(p + 1)}\text{4}\text{]n,…, [}\text{(p ? 1)}\text{2}\text{]n}$. The second, analogous to a theorem of Zolotareff and valid only if p ≡ 1 (mod 4), expresses ($\text{n}\text{p}$) in terms of the parity of the permutation of the set {$\text{1,2,…, (}\text{(p? 1)}\text{2}$} defined by the absolute values of the numerically least residues of $\text{n, 2n,…,[}\text{(p? 1}\text{2}\text{]n}$. 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.