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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**[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**, https://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. 108-109.**[3]**Daniel Shanks,*A sieve method for factoring numbers of the form 𝑛²+1*, Math. Tables Aids Comput.**13**(1959), 78–86. MR**0105784**, https://doi.org/10.1090/S0025-5718-1959-0105784-2**[4]**Daniel Shanks,*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272–284. MR**0108470**, https://doi.org/10.1090/S0025-5718-1959-0108470-8**[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 and E. M. Wright,*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[11]**Ernst Trost,*Primzahlen*, Verlag Birkhäuser, Basel-Stuttgart, 1953 (German). MR**0058630****[12]**Atle Selberg,*The general sieve-method 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**

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DOI:
https://doi.org/10.1090/S0025-5718-1960-0120203-6

Article copyright:
© Copyright 1960
American Mathematical Society