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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Euler’s criterion for cubic nonresidues
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by Kenneth S. Williams PDF
Proc. Amer. Math. Soc. 49 (1975), 277-283 Request permission

Abstract:

If $p$ is a prime $\equiv 1 \pmod 3$ there are integers $L$ and $M$ such that $4p = {L^2} + 27{M^2},L \equiv 1\pmod 3$. Indeed $L$ and ${M^2}$ are unique. If $D$ is a cubic nonresidue $\pmod p$ it is shown how to choose the sign of $M$ so that \[ {D^{(p - 1)/3}} \equiv (L + 9M)/(L - 9M)\pmod p.\] The case $D = 2$ has been treated by Emma Lehmer.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 49 (1975), 277-283
  • MSC: Primary 10A10
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0366792-0
  • MathSciNet review: 0366792