Diophantine determinations of 3^(p−1)∕8 and 5^(p−1)∕4

Richard Howard Hudson

Vol. 111 (1984), No. 1, 49–55

DOI: 10.2140/pjm.1984.111.49

Abstract

Let p be a prime = 24f + 1. The author and Kenneth S. Williams derived a criteria for 3 to be an eighth power (mod p) in terms of the parameters in the Diophantine systems a² + b² and x² + 3y². A new proof of this result is given which uses Jacobi sums. This proof is simpler in that it does not require summing 36 cyclotomic numbers; moreover, it leads simultaneously to new necessary and sufficient criteria for 3^(p−1)∕8 to be congruent to b∕a (mod p),a ≡ 1 (mod 4),b > 0. Using this result, criteria for 3^(p−1)∕8 ≡ 1,b∕a,−1, or −b∕a (mod p) are given in terms of the parameters in other well-known quadratic partitions of p or of 4p.

Let p be a prime = 20f + 1, 16p = x² + 50u² + 50v² + 125w², xw = v² − 4uv −u². It is shown that 5^(p−1)∕4 ≡ 1 (mod p) if and only if 16∣w or uv ≡ 2 (mod 4). This result is of interest in relation to criteria given by Emma Lehmer for 2 to be a fifth power (mod p) and for p to be a hyperartiad.

Mathematical Subject Classification 2000

Primary: 11A15

Secondary: 11L10

Milestones

Received: 9 December 1981

Published: 1 March 1984