The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit

TL;DR

This paper explores properties of initial numbers and repunit numbers, proving key gcd and divisibility properties, and analyzing their prime divisors, contributing to the theoretical understanding of these special numbers.

Contribution

It introduces new proofs of gcd and divisibility properties of repunit numbers and examines their prime divisor structure, advancing the theoretical framework of number theory.

Findings

gcd(R_a, R_b) = R_{gcd(a,b)}

R_{ab}/(R_a R_b) is an integer iff gcd(a,b)=1

Divisors of repunit numbers relate to prime number degrees

Abstract

In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved: $gcd(R_a, R_b) = R_{gcd(a,b)}$; $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\geq1$, $b\geq1$ are integers. Dividers of numbers repunit, are researched by a degree of prime number.