Divisibility properties of sum of powers of successive natural numbers and Bowen’s hypothesis

Article type: Research Article

Authors: Mestechkin, M.

Affiliations: Independent Scholar, 12773 Seabreeze Farms Dr. # 33, San Diego, CA, USA | E-mail: mestechkinmm@gmail.com

Correspondence: [*] Corresponding author: Independent Scholar, 12773 Seabreeze Farms Dr. # 33, San Diego, CA, USA. E-mail: mestechkinmm@gmail.com.

Abstract: The explicit formulae for remainder of sum of powers of successive natural numbers (SPSN) divided by any prime factor p of the number N of sum terms is derived. The remainder is zero, if N contains pj, j>1. It is also 0, if j=1 and the exponent is not divisible by p-1. The expression (N-N/p) (mod p), named “eigen-remainder” introduced for N is the remainder of SPSN in the rest of cases. All natural numbers, having all eigen-remainders equal to 1, are found not exceeding 42. It is demonstrated that in all other cases, there exists a prime divisor of N, which dividing SPSN gives the remainder ≠ 1. This proves Bowen’s hypothesis.

Keywords: Stirling numbers, Faulhaber formula, Chinese remainder theorem, Bowen’s hypothesis, averaged natural power

DOI: 10.3233/JCM-240001

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 24, no. 4-5, pp. 3333-3339, 2024