Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 31, 07 March 2024
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Research on Euler Totient function equation kφ(n)=n-1
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Abstract
Let φ denote Euler’s Totient function. There are some properties about φ(n), when n is a prime or n=p_1^(r_1 )⋯p_k^(r^k ). The Euler’s function equation, kφ(n)=n-1(1), where k is a positive integer, and n is a composite number, is called Lehmer’s conjecture. Lehmer mentioned a series of properties of n that satisfy the equation in his own thesis and provided some proof. Afterwards, Ke Zhao and Sun Qi conducted further research. In previous studies, this conjecture was considered correct, but it is difficult to prove it. The case k=2 has been discussed and proved that when k=2 and n=p_1 p_2,... p_i are different prime numbers. Also, some properties of the composite numbers that satisfy the equation have also been proven. Some conclusions can be proven, by using elementary number theory methods. Using these conclusions, we can conclue that when k=2, the solution of (1) is at least the product of 12 odd prime numbers.
Keywords
Number Theory, Euler Totient Function, Lehmer’s Conjecture
References
1. DH Lehmer. On euler’s totient function. 1932.
2. Godfrey Harold Hardy and Edward Maitland Wright. An introduction to the theory of numbers. Oxford university press, 1979.
3. K.Zhao and S.Qi. On equation kφ(n)=n-1. Journal of Sichuan University (Natural Science Edition), pages 13-21, 1963.
4. Florian Luca and Carl Pomerance. On composite integers n for which ϕ(n)∣n-1. Bol. Soc. Mat. Mexicana, 17(3):13-21, 2011.
5. G Tenenbaum. Cambridge stud. adv. math. 46, 1995.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-317-3
- ISBN (Online)
- 978-1-83558-318-0
- Published Date
- 07 March 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/31/20241069
- Copyright
- 07 March 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited