Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T23:24:01.376Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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