## Quadratic polynomials which have a high density of prime values

HTML articles powered by AMS MathViewer

- by G. W. Fung and H. C. Williams PDF
- Math. Comp.
**55**(1990), 345-353 Request permission

## Abstract:

The University of Manitoba Sieve Unit is used to find several values of $A ( > 0)$ such that the quadratic polynomial ${x^2} + x + A$ will have a large asymptotic density of prime values. The Hardy-Littlewood constants which characterize this density are also evaluated.## References

- H. Cohen and H. W. Lenstra Jr.,
*Heuristics on class groups of number fields*, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR**756082**, DOI 10.1007/BFb0099440 - Gary Cornell and Lawrence C. Washington,
*Class numbers of cyclotomic fields*, J. Number Theory**21**(1985), no. 3, 260–274. MR**814005**, DOI 10.1016/0022-314X(85)90055-1 - 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**, DOI 10.1007/BF02403921 - Edgar Karst,
*The congruence $2^{p-1}\equiv 1$ $(mod$ $p^{2})$ and quadratic forms with high density of primes*, Elem. Math.**22**(1967), 85–88. MR**215777**
D. H. Lehmer, - D. H. Lehmer, Emma Lehmer, and Daniel Shanks,
*Integer sequences having prescribed quadratic character*, Math. Comp.**24**(1970), 433–451. MR**271006**, DOI 10.1090/S0025-5718-1970-0271006-X - H. W. Lenstra Jr.,
*On the calculation of regulators and class numbers of quadratic fields*, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR**697260** - C. D. Patterson and H. C. Williams,
*A report on the University of Manitoba Sieve Unit*, Congr. Numer.**37**(1983), 85–98. MR**703580** - Daniel Shanks,
*A sieve method for factoring numbers of the form $n^{2}+1$*, Math. Tables Aids Comput.**13**(1959), 78–86. MR**105784**, DOI 10.1090/S0025-5718-1959-0105784-2 - Daniel Shanks,
*On the conjecture of Hardy & Littlewood concerning the number of primes of the form $n^{2}+a$*, Math. Comp.**14**(1960), 320–332. MR**120203**, DOI 10.1090/S0025-5718-1960-0120203-6 - Daniel Shanks,
*Supplementary data and remarks concerning a Hardy-Littlewood conjecture*, Math. Comp.**17**(1963), 188–193. MR**159797**, DOI 10.1090/S0025-5718-1963-0159797-6 - Daniel Shanks,
*Class number, a theory of factorization, and genera*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR**0316385**
—, - Daniel Shanks,
*Calculation and applications of Epstein zeta functions*, Math. Comp.**29**(1975), 271–287. MR**409357**, DOI 10.1090/S0025-5718-1975-0409357-2
—,

*On the function of*${x^2} + x + A$, Sphinx

**6**(1936), 212-214 and Sphinx

**7**(1937), 40.

*Systematic examination of Littlewood’s bounds on*$L(1,\chi )$, Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, R.I., 1973, pp. 267-283.

*A survey of quadratic, cubic and quartic algebraic number fields*, Congr. Numer.

**17**(1976), 15-42.

## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp.
**55**(1990), 345-353 - MSC: Primary 11N32; Secondary 11-04, 11Y35
- DOI: https://doi.org/10.1090/S0025-5718-1990-1023759-3
- MathSciNet review: 1023759