On the conjecture of Hardy & Littlewood concerning the number of primes of the form

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 | Similar Articles | Additional Information

**[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. 108-109.**[3]**Daniel Shanks, ``A sieve method for factoring numbers of the form ,''*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 ,''*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: , , , , 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. 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