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

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**14**(1960), 321-332 Request permission

## References

- G. H. Hardy and J. E. Littlewood,
*Some problems of âPartitio numerorumâ; III: On the expression of a number as a sum of primes*, Acta Math.**44**(1923), no.Â 1, 1â70. MR**1555183**, DOI 10.1007/BF02403921
A. E. Western, âNote on the number of primes of the form ${n^2} + 1$,â Cambridge Phil. Soc., - Daniel Shanks,
*A sieve method for factoring numbers of the form $n^{2}+1$*, Math. Tables Aids Comput.**13**(1959), 78â86. MR**105784**, DOI 10.1090/S0025-5718-1959-0105784-2 - Daniel Shanks,
*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272â284. MR**108470**, DOI 10.1090/S0025-5718-1959-0108470-8
Daniel Shanks, âOn the conjecture of Hardy and Littlewood concerning the number of primes of the form ${n^2} + a$,â - G. H. Hardy and E. M. Wright,
*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125** - Ernst Trost,
*Primzahlen*, Verlag BirkhĂ¤user, Basel-Stuttgart, 1953 (German). MR**0058630** - 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.Â 286â292. MR**0044563**

*Proc.*, v. 21, 1922, p. 108-109.

*Notices*, Amer. Math. Soc., v. 6, 1959, p. 417. Abstract 559-52. The numbers

*b*(

*s*) also arise in an entirely different connectionâthey are related to the number of distinct

*circular parity switches of order s*. See Daniel Shanks, âA circular parity switch and applications to number theory,â

*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 âlittleâ theoremâsee (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). E. Landau,

*Aus der elementaren Zahlentheorie*, Chelsea, 1946, Part IV, Chap. 6-9. 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 ${L_{ - 3}}(s) = {t_n}$. A similar sieve argument was given for the twin prime problem in Charles S. Sutton, âAn investigation of the average distribution of twin prime numbers,â

*Jn. Math. Phys.*, v. 16, 1937, p. 1-42.

## Additional Information

- © Copyright 1960 American Mathematical Society
- 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