On the conjecture of Hardy & Littlewood concerning the number of primes of the form
Author:
Daniel Shanks
Journal:
Math. Comp. 14 (1960), 321332
MSC:
Primary 10.00
MathSciNet review:
0120203
Fulltext PDF Free Access
References 
Similar Articles 
Additional Information
 [1]
G.
H. Hardy and J.
E. Littlewood, Some problems of ‘Partitio numerorum’;
III: On the expression of a number as a sum of primes, Acta Math.
44 (1923), no. 1, 1–70. MR
1555183, http://dx.doi.org/10.1007/BF02403921
 [2]
A. E. Western, ``Note on the number of primes of the form ,'' Cambridge Phil. Soc., Proc., v. 21, 1922, p. 108109.
 [3]
Daniel
Shanks, A sieve method for factoring numbers
of the form 𝑛²+1, Math. Tables
Aids Comput. 13
(1959), 78–86. MR 0105784
(21 #4520), http://dx.doi.org/10.1090/S00255718195901057842
 [4]
Daniel
Shanks, Quadratic residues and the
distribution of primes, Math. Tables Aids
Comput. 13 (1959),
272–284. MR 0108470
(21 #7186), http://dx.doi.org/10.1090/S00255718195901084708
 [5]
Daniel Shanks, ``On the conjecture of Hardy and Littlewood concerning the number of primes of the form ,'' Notices, Amer. Math. Soc., v. 6, 1959, p. 417. Abstract 55952.
 [6]
The numbers b(s) also arise in an entirely different connectionthey 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 5437. It was in this connection that the author first noted the unusual proof of a special case of the Fermat ``little'' theoremsee (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. 69.
 [8]
Fletcher, Miller & Rosenhead, Index of Mathematical Tables, McGrawHill, 1946, p. 42, 43, p. 63. The correspondence between our notation and theirs is as follows: , , , , and .
 [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. 142.
 [10]
G.
H. Hardy and E.
M. Wright, An introduction to the theory of numbers, Oxford,
at the Clarendon Press, 1954. 3rd ed. MR 0067125
(16,673c)
 [11]
Ernst
Trost, Primzahlen, Verlag Birkhäuser, BaselStuttgart,
1953 (German). MR 0058630
(15,401g)
 [12]
Atle
Selberg, The general sievemethod and its place in prime number
theory, Proceedings of the International Congress of Mathematicians,
Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, R. I., 1952,
pp. 286–292. MR 0044563
(13,438d)
 [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 ,'' Cambridge Phil. Soc., Proc., v. 21, 1922, p. 108109.
 [3]
 Daniel Shanks, ``A sieve method for factoring numbers of the form ,'' MTAC, v. 13, 1959, p. 7886. MR 0105784 (21:4520)
 [4]
 Daniel Shanks, ``Quadratic residues and the distribution of primes,'' MTAC, v. 13, 1959, p. 272284. MR 0108470 (21:7186)
 [5]
 Daniel Shanks, ``On the conjecture of Hardy and Littlewood concerning the number of primes of the form ,'' Notices, Amer. Math. Soc., v. 6, 1959, p. 417. Abstract 55952.
 [6]
 The numbers b(s) also arise in an entirely different connectionthey 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 5437. It was in this connection that the author first noted the unusual proof of a special case of the Fermat ``little'' theoremsee (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. 69.
 [8]
 Fletcher, Miller & Rosenhead, Index of Mathematical Tables, McGrawHill, 1946, p. 42, 43, p. 63. The correspondence between our notation and theirs is as follows: , , , , and .
 [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. 142.
 [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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718196001202036
PII:
S 00255718(1960)01202036
Article copyright:
© Copyright 1960 American Mathematical Society
