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On the quintic character of 2

  • Helen Popova Alderson (a1)
Abstract

Criteria for 2 to be an e-th power residue of a prime

p ≡ 1 (mod e = ef+1,

have been obtained in various forms for e = 2, 3, 5. Euler proved the well known result that 2 is a quadratic residue of a prime p ≡ ± l (mod 8). Dickson [1] showed that 2 is a cubic residue of p ≡ 1 (mod 3) if and only if p = L2+27M2 is soluble in integers L, M.

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References
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1. Dickson L. E., “Cyclotomy, higher congruences and Waring's problem”, American J. of Math., 57 (1935), 391424.
2. Lehmer Emma, “The quintic character of 2 and 3”, Duke Math. J., 18 (1951), 1118.
3. Neil Y. Wilson, “Conjectures as to a factor of 2p+1”, Edinburgh Math. Notes, 39 (1954), 69.
4. Euler , Correspondance mathématique et physique (St. Petersbourg, 1), 1843, 146149.
5. Lehmer Emma, “On Euler's criterion”, Mathematika, 1 (1954), 6470.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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