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On the quintic character of 2

Published online by Cambridge University Press:  26 February 2010

Helen Popova Alderson
Helen Popova Alderson
Affiliation:
7, Highfield Avenue, Cambridge.
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Extract

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.

Type
Research Article
Information
Mathematika , Volume 11 , Issue 2 , December 1964 , pp. 125 - 130
Copyright
Copyright © University College London 1964

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References

1. Dickson, L. E., “Cyclotomy, higher congruences and Waring's problem”, American J. of Math., 57 (1935), 391–424.CrossRefGoogle Scholar
2. Lehmer, Emma, “The quintic character of 2 and 3”, Duke Math. J., 18 (1951), 11–18.CrossRefGoogle Scholar
3. Neil, Y. Wilson, “Conjectures as to a factor of 2p+1”, Edinburgh Math. Notes, 39 (1954), 6–9.Google Scholar
4. Euler, , Correspondance mathématique et physique (St. Petersbourg, 1), 1843, 146–149.Google Scholar
5. Lehmer, Emma, “On Euler's criterion”, Mathematika, 1 (1954), 64–70.Google Scholar