Abstract
For integers a and b we define the Shanks chain p 1, p 2,..., p k of length k to be a sequence of k primes such that \(p_{i+1} = ap_{i}^{2} -- b\) for i = 1,2,..., k − 1. While for Cunningham chains it is conjectured that arbitrarily long chains exist, this is, in general, not true for Shanks chains. In fact, with s = ab we show that for all but 56 values of s ≤1000 any corresponding Shanks chain must have bounded length. For this, we study certain properties of functional digraphs of quadratic functions over prime fields, both in theory and practice. We give efficient algorithms to investigate these properties and present a selection of our experimental results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bach, E.: Toward a theory of Pollard’s rho method. Information and Computation 90, 139–155 (1991)
Bach, E., Shallit, J.O.: Algorithmic number theory, vol. 1. MIT Press, Cambridge (1996)
Barrucand, P., Cohn, H.: A note on primes of type x2 + 32y2, class number and residuacity. J. Reine Angew. Math. 238, 67–70 (1969)
Bateman, P.T., Horn, R.: Primes represented by irreducible polynomials in one variable. In: Proc. Sympos. Pure Math., Providence, AMS, vol. VIII, pp. 119–132(1965)
Dickson, E.: History of the theory of numbers, vol. 1. Chelsea, New York (1952)
Escott, E.B.: Cubic congruences with three real roots. Annals of Math. 11(2), 86–92 (1909–1910)
Flajolet, P., Odlyzko, A.M.: Random mapping statistics. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 329–354. Springer, Heidelberg (1990)
Forbes, T.: Prime clusters and Cunningham chains. Math. Comp. 68(228), 1739–1747 (1999)
Guy, R.K.: Unsolved problems in number theory, 2nd edn. Springer, Berlin (1994)
Hardy, G.H., Littlewood, J.E.: Some problems of partitio numerorum. III. On the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1923)
LiDIA Group, Technische Universität Darmstadt, Darmstadt, Germany, LiDIA - a library for computational number theory, version 1.3 (1997)
Morton, P.: Arithmetic properties of periodic points of quadratic maps. Acta Arithmetica 62, 343–372 (1992)
Morton, P.: Arithmetic properties of periodic points of quadratic maps II. Acta Arithmetica 87, 89–102 (1998)
Narkiewicz, W.: Polynomial cycles in algebraic number fields. Colloq. Math. 58, 151–155 (1989)
Pollard, J.M.: A Monte Carlo method for factorization. BIT 15(3), 331–335 (1975)
Schinzel, A., Sierpiński, W.: Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4, 185–208 (1958)
Schinzel, A., Sierpiński, W.: Remarks on the paper “Sur certaines hypothèses concernant les nombres premiers”. Acta Arith. 7, 1–8 (1961)
Shanks, D.: Letter to D.H. and Emma Lehmer, June 10 (1969)
Shanks, D.: Supplemental data and remarks concerning a Hardy-Littlewood conjec ture. Math. Comp. 17, 188–193 (1963)
Stein, A., Williams, H.: An improved method of computing the regulator of a real quadratic function field. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 607–620. Springer, Heidelberg (1998)
Williams, H.C., Holte, R.: Computation of the solution x3 + dy3 = 1. Math. Comp. 31, 778–785 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Teske, E., Williams, H.C. (2000). A Note on Shanks’s Chains of Primes. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_38
Download citation
DOI: https://doi.org/10.1007/10722028_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
Online ISBN: 978-3-540-44994-2
eBook Packages: Springer Book Archive