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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22319 |
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Table of 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 $B$.