Nagell’s totient revisited

Pentti Haukkanen and R. Sivaramakrishnan
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 4, 1998, Number 3, Pages 93—100
Download full paper: PDF, 320 Kb

Details

Authors and affiliations

Pentti Haukkanen
Department of Mathematical Sciences, University of Tampere,
P.O.Box 607, FIN-33101 Tampere, Finland

R. Sivaramakrishnan
Department of Studies in Mathematics, Mangalore University,
Dk 574199, India

Abstract

Nagell’s totient θ(n, r) counts the number of solutions of the congruence (*) n = x + y (mod r ) under the restriction (x, r) = (y, r) = 1. In this paper we evaluate the number θ(n, r, q) of solutions of the congruence (*) under the restriction (x,r) = (y,r) = q, where q|r, via Ramanathan’s approach to class-division of integers (mod r).

References

  1. Tom M. Apostol: Introduction to Analytic Number Theory, Springer-Verlag UTM (1976).
  2. Umberto Cerruti: Computing the number of restricted solutions of linear congruences by using generalized Ramanujan sums and matrices (Extended Abstract) (1996).
  3. Eckford Cohen: A class of arithmetical functions, Proc. Nat. Acad. Sci. (USA) 41 (1955), 939-944.
  4. P. Haukkanen and Paul J. McCarthy: Sums of values of even functions, Portugal. Math. 48 (1991), 53-66.
  5. Paul J. McCarthy: Counting restricted solutions of a linear congruence, Nieuw Arch. Wish. (3) XXV (1977), 133-147.
  6. Paul J. McCarthy: Introduction to Arithmetical Functions, Springer-Verlag Universitext (1986).
  7. T. Nagell: Verallgemeinerung eines Satzes von Schemmel, Skr. Norske Vod. Akad. Oslo (Math. Class) I, No. 13 (1923), 23-25.
  8. C. A. Nicol and H. S. Vandiver: A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Nat. Acad. Sci. (USA) 40 (1954), 825-835.
  9. K. G. Ramanathan: Some applications of Ramanujan’s trigonometrical sum Cm(n), Proc. Ind. Acad. Sci. (A) 20 (1944), 62-69.
  10. David Rearick: A linear congruence with side conditions, Amer. Math. Monthly 70 (1963), 837-840.
  11. R. Sivaramakrishnan: Classical Theory of Arithmetic Functions, Marcel Dekker: Monographs and Text Books in Pure and Applied Mathematics No. 126 (1989).
  12. R. Vaidyanathaswamy: A remarkable property of integers (mod N) and its bearing on group theory, Proc. Ind. Acad. Sci. Section A (1937), 63-75.

Related papers

Cite this paper

Haukkanen, P. and Sivaramakrishnan, R. (1998). Nagell’s totient revisited. Notes on Number Theory and Discrete Mathematics, 4(3), 93-100.

Comments are closed.