Persistence of form and the value group of reducible cubics

Elliott, P. D. T. A.

doi:10.1090/S0002-9947-1987-0869403-3

Persistence of form and the value group of reducible cubics
HTML articles powered by AMS MathViewer

by P. D. T. A. Elliott PDF

Trans. Amer. Math. Soc. 299 (1987), 133-143 Request permission

Abstract:

It is proved that the values of $x({x^2} + c)$, $c \ne 0$, at positive integers, multiplicatively generate the positive rationals. Analogs in rational function fields are obtained.

References

D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4 (1957), 106–112. MR 93504, DOI 10.1112/S0025579300001157
D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. (3) 12 (1962), 179–192. MR 132732, DOI 10.1112/plms/s3-12.1.179
D. A. Burgess, Dirichlet characters and polynomials, Proceedings of the International Conference Number Theory (Moscow, 1971), 1973, pp. 203–205, 266 (English, with Russian summary). MR 0401673
P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 272, Springer-Verlag, New York, 1985. MR 766558, DOI 10.1007/978-1-4613-8548-6
P. D. T. A. Elliott, The value distribution of reducible cubics, Canad. Math. Bull. 28 (1985), no. 3, 328–336. MR 790954, DOI 10.4153/CMB-1985-039-8
P. D. T. A. Elliott, On representing integers as products of integers of a prescribed type, J. Austral. Math. Soc. Ser. A 35 (1983), no. 2, 143–161. MR 704421
P. D. T. A. Elliott, On the mean value of $f(p)$, Proc. London Math. Soc. (3) 21 (1970), 28–96. MR 266881, DOI 10.1112/plms/s3-21.1.28
R. C. Vaughan, The Hardy-Littlewood method, Cambridge Tracts in Mathematics, vol. 80, Cambridge University Press, Cambridge-New York, 1981. MR 628618