Vol. 111, No. 1, 1984

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Diophantine determinations of 3(p1)8 and 5(p1)4

Richard Howard Hudson

Vol. 111 (1984), No. 1, 49–55

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(p1)8 to be congruent to b∕a (mod p),a 1 (mod 4),b > 0. Using this result, criteria for 3(p1)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(p1)4 1 (mod p) if and only if 16w 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
Received: 9 December 1981
Published: 1 March 1984
Richard Howard Hudson