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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.
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