Amicable numbers and their connection to the Euler totient function
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908734147526656 |
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| author | Mavaddat, Ali Reza Alikhani, Saeid |
| author_facet | Mavaddat, Ali Reza Alikhani, Saeid |
| contents | A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest common divisor is a power of two, using their distinct prime factorizations. Specifically, we examine pairs of the forms $A=2^n ab, B=2^n cd$, $A=2^n abc, B=2^n de$, and $A=2^n abc, B=2^n def$. From these configurations, we establish explicit symmetric identities that relate the sum $φ(A)+φ(B)$ of Euler's totient functions directly to the odd prime factors of $A$ and $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Amicable numbers and their connection to the Euler totient function Mavaddat, Ali Reza Alikhani, Saeid History and Overview 11A25, 11N25 A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest common divisor is a power of two, using their distinct prime factorizations. Specifically, we examine pairs of the forms $A=2^n ab, B=2^n cd$, $A=2^n abc, B=2^n de$, and $A=2^n abc, B=2^n def$. From these configurations, we establish explicit symmetric identities that relate the sum $φ(A)+φ(B)$ of Euler's totient functions directly to the odd prime factors of $A$ and $B$. |
| title | Amicable numbers and their connection to the Euler totient function |
| topic | History and Overview 11A25, 11N25 |
| url | https://arxiv.org/abs/2512.22319 |