A walk through the repunit sequence

Abstract

In this work, we consider a sequence formed only by repunit numbers R_n, where R_n denotes the numbers formed by repeating the digit 1. This approach has related to the properties of Lucas sequences, as discussed by Jaroma (2007). We highlight that for the numerical sequence of repunit numbers, the Catalan and Cassini identities also hold. In 1978, Yates claimed that there is a fascination with repunit numbers, which stems from their application in various recreational math problems. Here, we also exhibit some inherent properties of the numerical class of R_n, show some relationships between repunit numbers and powers of repunit numbers with natural exponents; furthermore, we study the divisibility relationship among its terms, particularly the prime factor characteristic of repunit numbers. Additionally, we prove the conjecture proposed by Costa and Santos (2022) regarding the quotient of a type of repunit number.