Amicable numbers and their connection to the Euler totient function

2025/12/26 by Ali Reza Mavaddat, Mavaddat, Ali Reza, Saeid Alikhani +1 · 1 voice
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Algebraic Geometry and Number Theory #math.HO

paper · pdf · doi:10.48550/arxiv.2512.22319

Abstract

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=2ⁿ ab, B=2ⁿ cd, A=2ⁿ abc, B=2ⁿ de, and A=2ⁿ abc, B=2ⁿ 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.

Discussions

• Ali Reza Mavaddat, Saeid Alikhani: Amicable numbers and their connection to the Euler totient function https://arxiv.org/abs/2512.22319 https://arxiv.org/pdf/2512.22319 https://arxiv.org/html/2512.223 [bsky, 0 points, 0 comments]

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