# 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.

## Key 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.