$d$-Translated Unit Sensitive Primes

Abstract

A nonnegative integer $n$ is $d$-translated unit sensitive when appending $d$ zeros to the right of the number and changing the unit digit to any possible unit digit (so long as the resulting number is not $n$) results in a composite number. We find an arithmetic progression of nonnegative integers that are $d$-translated unit sensitive for any nonnegative integer $d$. We construct the arithmetic progression such that there exists $k$ consecutive primes in the progression for any positive integer $k$. The first known prime that is $d$-translated unit sensitive for any nonnegative integer $d$ is found as a consequence. We also refine the arithmetic progression such that the integers in the progression are $d$-translated unit sensitive for any nonnegative integer $d$ and are Brier numbers. The refined arithmetic progression contains $k$ consecutive primes for any choice of positive integer $k$.