The septic character of \(2, 3, 5\) and \(7\). (English) Zbl 0265.10004

Leonard, Philip A.; Williams, Kenneth S.

Pac. J. Math. 52, 143-147 (1974).

Necessary and sufficient conditions for \(2,3,5\) and \(7\) to be seventh powers \(\pmod p\) (\(p\) a prime \(\equiv 1\pmod 7\) are determined in terms of the solutions of the triple of Diophantine equations
\[ 72p = 2x_1^2+ 42(x_2^2 +x_3^2+ x_4^2) + 343(x_5^2+ 3x_6^2), \]
\[ 12x_2^2- 12x_4^2 +147x_5^2-441x_6^2 + 56x_1x_6 +24x_2x_3 - 24x_2x_4 + 48x_3x_4 + 98x_5x_6 = 0, \]
\[ 12x_3^2 -12x_4^2 +49x_5^2 -147x_6^2+28x_1x_5+28x_1x_6+ 48x_2x_3 +24x_2x_4 +24x_3x_4 +490x_5x_6 =0, \]
\(x_1\equiv 1\pmod 7\).

Reviewer: Kenneth S. Williams (Ottawa)

MSC:

11A15	Power residues, reciprocity
11T22	Cyclotomy

Keywords:

seventh powers modulo a prime; septic character