A Theorem on the Function Ω(n) and Mersenne Numbers

We prove that for every positive integer n, the number of prime factors of 2ⁿ−1 (counted with multiplicity) is greater than or equal to the number of prime factors of n. The proof is based on an original form of induction on Ω(n), the total number of prime divisors of n. 📄 Full text available at DOI: https://doi.org/10.5281/zenodo.16447827

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2025-07-25
DOI
https://doi.org/10.5281/zenodo.16447827
Primary Topic
Genetics and Neurodevelopmental Disorders
Type
article
Field-Weighted Citation Impact
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article

A Theorem on the Function Ω(n) and Mersenne Numbers

Zenodo (CERN European Organization for Nuclear Research)
Genetics and Neurodevelopmental Disorders
article

A Theorem on the Function Ω(n) and Mersenne Numbers

article en

Abstract

We prove that for every positive integer n, the number of prime factors of 2ⁿ−1 (counted with multiplicity) is greater than or equal to the number of prime factors of n. The proof is based on an original form of induction on Ω(n), the total number of prime divisors of n. 📄 Full text available at DOI: https://doi.org/10.5281/zenodo.16447827

Zenodo (CERN European Organization for Nuclear Research)
Openalex Percentile: Top 50%
Genetics and Neurodevelopmental Disorders
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A Theorem on the Function Ω(n) and Mersenne Numbers · Zenodo (CERN European Organization for Nuclear Research) (2025) | TGRS Research Map | TGRS