Carmichael’s conjecture and a minimal unique solution

W. Ramadan-Jradi
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 5, 1999, Number 2, Pages 55—70
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Authors and affiliations

W. Ramadan-Jradi
School of Mathematical Sciences, University of Technology,
Sydney, P.O. Box 123, Broadway, NSW 2007, Australia


R. D. Carmichael [1] conjectured that: “The equation ϕ(x) = A where ϕ is Euler’s totient function, and A is an even positive integer does not have a unique solution. This paper is a continue of paper [5], and it states some theorems and lemmas which help giving an “if and only if” condition for a unique solution, and locate this solution when it exists. By following the same approach of [5] we will be able to determine some particular properties of the set FA(k+1) , where FA(k) = {x : ϕ(x) = A(k)} as defined in [5] is a minimal set of solutions of the equation ϕ(x) = A(k).

AMS Classification

  • 11A25


  1. R. D. Carmichael, Note on Euler’s ϕ-function, Bull. Amer. Math. Soc. 28 (1922), 109-110.
  2. P. Hagis, On Carmichael’s Conjecture concerning the Euler Phi- Function, Bollettino U. M. I (6) 5-A (1986), 409-412.
  3. V. L. Klee, On a Conjecture of Carmichael, American Mathematical Society. Bulletin. 53 (1947), 1183-1186.
  4. N. S. Mendelsohn, The equation ϕ(x) = k. Mathematics Magazine 49, (1) (1976), 37-39.
  5. W.A. Ramadan-Jradi, Some Constraints On Carmichael’s Conjecture, preprint (1997).
  6. A. Schlafly & S. Wagon, Carmichael’s Conjecture on the Euler function is valid below 1010,000,000, Mathematics of Computation. 63,(207) (1994), 415-419.

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Cite this paper

Ramadan-Jradi, W. (1999). Carmichael’s conjecture and a minimal unique solution. Notes on Number Theory and Discrete Mathematics, 5(2), 55-70.

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