arXiv subjects

Thomas Luckner

Publications and source records attributed to Thomas Luckner.

4 recordsLinked to original sources

arXiv2026

$d$-Translated Unit Sensitive Primes

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 $k$.

math.NT↗

arXiv2023

Generalized Sierpi\'nski Numbers

A Sierpi\'nski number is a positive odd integer $k$ such that $k \cdot 2^n + 1$ is composite for all positive integers $n$. Fix an integer $A$ with $2 \le A$. We show that there exists a positive odd integer $k$ such that $k\cdot a^n + 1$ is composite for all integers $a \in [2, A]$ and all $n \in \mathbb{Z}^+$.

math.NT↗

arXiv2023

On $n^{\rm th}$ order Euler polynomials of degree $n$ that are Eisenstein

For $m$ an even positive integer and $p$ a prime, we show that the generalized Euler polynomial $E_{mp}^{(mp)}(x)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m (2^m-1)B_m$. As a consequence, we deduce that at least $1/3$ of the generalized Euler polynomials $E_n^{(n)}(x)$ are in Eisenstein form with respect to a prime $p$ dividing $n$ and, hence, irreducible over $\mathbb Q$.

math.NT↗

arXiv2022

Consecutive primes which are widely digitally delicate and Brier numbers

Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer $k$, there exist $k$ consecutive primes which are Brier numbers. We show that for every positive integer $k$, there exist $k$ consecutive primes that are both widely digitally delicate and Brier numbers.

math.NT↗