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- Title
The non-trivial solutions of the arithmetic functional equation φ(x) =S(x<sup>k</sup>).
- Authors
Xiao-wei, PAN
- Abstract
For any positive integer n, let φ(n) and S(n) denote the Euler function and the Smarandache function respectively. For a fixed positive integer k, if x is a positive integer satisfying x> 1 and φ(x) = S (xk), then x is called a non-trivial solution of the equation φ(x)=S(xk). Using some elementary number theory methods, it is proved that (i) All non-trivial solutions x of φ(x)=S(xk) satisfy 2k<x≤64k³; (ii) There exists infinitely many k which make φ(x)=S(xk) has at least two non-trivial solutions.
- Subjects
ARITHMETIC functions; NUMERICAL solutions to functional equations; EULER method; SMARANDACHE function; INTEGERS
- Publication
Basic Sciences Journal of Textile Universities / Fangzhi Gaoxiao Jichu Kexue Xuebao, 2012, Vol 25, Issue 3, p339
- ISSN
1006-8341
- Publication type
Article