W. Ramadan-Jradi

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 5, 1999, Number 2, Pages 55–70

**Full paper (PDF, 385 Kb)**

## Details

### Authors and affiliations

W. Ramadan-Jradi

*School of Mathematical Sciences, University of Technology,
Sydney, P.O. Box 123, Broadway, NSW 2007, Australia*

### Abstract

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 *F*_{A(k+1)} , where *F*_{A(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

### References

- R. D. Carmichael, Note on Euler’s
*ϕ*-function, Bull. Amer. Math. Soc. 28 (1922), 109-110. - P. Hagis, On Carmichael’s Conjecture concerning the Euler Phi- Function, Bollettino U. M. I (6) 5-A (1986), 409-412.
- V. L. Klee, On a Conjecture of Carmichael, American Mathematical Society. Bulletin. 53 (1947), 1183-1186.
- N. S. Mendelsohn, The equation
*ϕ*(*x*) =*k*. Mathematics Magazine 49, (1) (1976), 37-39. - W.A. Ramadan-Jradi, Some Constraints On Carmichael’s Conjecture, preprint (1997).
- A. Schlafly & S. Wagon, Carmichael’s Conjecture on the Euler function is valid below 10
^{10,000,000}, Mathematics of Computation. 63,(207) (1994), 415-419.

## Related papers

## 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.