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The septic character of $$2, 3, 5$$ and $$7$$. (English) Zbl 0265.10004
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
$72p = 2x_1^2+ 42(x_2^2 +x_3^2+ x_4^2) + 343(x_5^2+ 3x_6^2),$
$12x_2^2- 12x_4^2 +147x_5^2-441x_6^2 + 56x_1x_6 +24x_2x_3 - 24x_2x_4 + 48x_3x_4 + 98x_5x_6 = 0,$
$12x_3^2 -12x_4^2 +49x_5^2 -147x_6^2+28x_1x_5+28x_1x_6+ 48x_2x_3 +24x_2x_4 +24x_3x_4 +490x_5x_6 =0,$ $$x_1\equiv 1\pmod 7$$.

##### MSC:
 11A15 Power residues, reciprocity 11T22 Cyclotomy
##### Keywords:
seventh powers modulo a prime; septic character
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