# Repeatedly Appending Any Digit to Generate Composite Numbers

## TL;DR

This paper explores the existence of infinitely many integers for which appending any number of a specific digit results in a composite number, revealing new insights into number construction and digit appending properties.

## Contribution

It proves the existence of infinitely many integers coprime to all digits that remain composite when any digit is appended repeatedly.

## Key findings

- Existence of infinitely many such integers proven.
- These integers are coprime to all digits.
- Appending any digit repeatedly yields composite numbers.

## Abstract

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending {\it any} digit yields a composite number.