On the Radical of Multiperfect Numbers and Applications

2018/12/28 by Nithin Kavi, Kavi, Nithin, Xinyi Zhang +5
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1812.11239

openalex publication_date 2018/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is conjectured that for a perfect number m, \rmrad(m)≪ m^(1)/(2). We prove bounds on the radical of multiperfect number m depending on its abundancy index. Assuming the ABC conjecture, we apply this result to study gaps between multiperfect numbers, multiperfect numbers represented by polynomials. Finally, we prove that there are only finitely many multiperfect multirepdigit numbers in any base g where the number of digits in the repdigit is a power of 2. This generalizes previous works of several authors including O. Klurman, F. Luca, P. Polack, C. Pomerance and others.

Citations

• On the distance between perfect numbers

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