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On the conjecture of Hardy & Littlewood concerning the number of primes of the form $ n\sp{2}+a$


Author: Daniel Shanks
Journal: Math. Comp. 14 (1960), 321-332
MSC: Primary 10.00
DOI: https://doi.org/10.1090/S0025-5718-1960-0120203-6
MathSciNet review: 0120203
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References [Enhancements On Off] (What's this?)

  • [1] G. H. Hardy & J. E. Littlewood, ``Partitio numerorum III: On the expression of a number as a sum of primes,'' Acta Math., v. 44, 1923, p. 48. MR 1555183
  • [2] A. E. Western, ``Note on the number of primes of the form $ {n^2} + 1$,'' Cambridge Phil. Soc., Proc., v. 21, 1922, p. 108-109.
  • [3] Daniel Shanks, ``A sieve method for factoring numbers of the form $ {n^2} + 1$,'' MTAC, v. 13, 1959, p. 78-86. MR 0105784 (21:4520)
  • [4] Daniel Shanks, ``Quadratic residues and the distribution of primes,'' MTAC, v. 13, 1959, p. 272-284. MR 0108470 (21:7186)
  • [5] Daniel Shanks, ``On the conjecture of Hardy and Littlewood concerning the number of primes of the form $ {n^2} + a$,'' Notices, Amer. Math. Soc., v. 6, 1959, p. 417. Abstract 559-52.
  • [6] The numbers b(s) also arise in an entirely different connection--they are related to the number of distinct circular parity switches of order s. See Daniel Shanks, ``A circular parity switch and applications to number theory,'' Notices, Amer. Math. Soc., v. 5, 1958, p. 96. Abstract 543-7. It was in this connection that the author first noted the unusual proof of a special case of the Fermat ``little'' theorem--see (9a) above. Likewise it was in this connection that Bernard Elpsas, in a private communication to the author (Sept. 3, 1958), developed the formula (9).
  • [7] E. Landau, Aus der elementaren Zahlentheorie, Chelsea, 1946, Part IV, Chap. 6-9.
  • [8] Fletcher, Miller & Rosenhead, Index of Mathematical Tables, McGraw-Hill, 1946, p. 42, 43, p. 63. The correspondence between our notation and theirs is as follows: $ {L_1}(s) = {u_n}$, $ {L_2}(s) = {p_n}$, $ {L_{ - 2}}(s) = {q_n}$, $ {L_3}(s) = {h_n}$, and $ {L_{ - 3}}(s) = {t_n}$.
  • [9] A similar sieve argument was given for the twin prime problem in Charles S. Sutton, ``An investigation of the average distribution of twin prime numbers,'' Jn. Math. Phys., v. 16, 1937, p. 1-42.
  • [10] G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, Oxford, 1938, p. 349. MR 0067125 (16:673c)
  • [11] Ernst Trost, Primzahlen, Basel, 1953, Chap. IX. MR 0058630 (15:401g)
  • [12] A. Selberg, ``The general sieve method and its place in prime number theory,'' Proc., Inter. Congress Math., Cambridge, 1950, p. 286. MR 0044563 (13:438d)

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DOI: https://doi.org/10.1090/S0025-5718-1960-0120203-6
Article copyright: © Copyright 1960 American Mathematical Society

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