Quadratic residues and quartic residues modulo primes

Quartic Residues and Binary Quadratic Forms

Let p ≡ 1 (mod 4) be a prime, m ∈ Z and p m. In this paper we obtain a general criterion for m to be a quartic residue (mod p) in terms of appropriate binary quadratic forms. Let d > 1 be a squarefree integer such that ( d p ) = 1, where ( d p ) is the Legendre symbol, and let εd be the fundamental unit of the quadratic field Q( √ d). Since 1942 many mathematicians tried to characterize those p...

Note on an Additive Characterization of Quadratic Residues Modulo p

It is shown that an even partition A∪B of the set R = {1, 2, . . . , p− 1} of positive residues modulo an odd prime p is the partition into quadratic residues and quadratic non-residues if and only if the elements of A and B satisfy certain additive properties, thus providing a purely additive characterization of the set of quadratic residues. 1 Additive properties of quadratic residues An inte...

Distribution of Residues Modulo p

The distribution of quadratic residues and non-residues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it is still an open problem to find a small upper bound on the least quadratic non-residue mod p as a fu...

On quadratic non-residues

The Distribution of Quadratic Residues and Non-residues

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality ...