Showing papers on "Quintic function published in 1974"

On the quartic character of quadratic units.

Abstract: In recent years a number of authors [1], [2], [3], [4], [7], [8], [9] have dealt from various points of view with the question of whether a fundamental unit sq in the quadratic field Q (j/g) is a quadratic residue of a prime p of which g is a quadratic residue. In this paper we will address ourselves to the following problem: Given that g and £0, where g is a prime, are quadratic residues of a prime jt>, find the conditions for sq to be a quartic residue of p. Furthermore, if these conditions are satisfied determine, if possible, whether it is an octic residue of p. The answers to these questions are in terms of the representation of p by binary quadratic forms and have applications to tests for primality and to the divisibility by p of certain terms of the corresponding Lucas' series. These will be discussed in sections 3 and 4. § 1. Besults and conjeetures We begin with q = 2 and p = 8n + l, so that ε2 = ί + J/2 is a rational integer modulo p. Every such prime p is uniquely represented by (1) p = a + 16fc = c + 8d (a Ξ c s l (mod 4)). It was shown in [1] that