# Truncatable primes and unavoidable sets of divisors

• Volume: 14, Issue: 1, page 21-25
• ISSN: 1804-1388

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## Abstract

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We are interested whether there is a nonnegative integer ${u}_{0}$ and an infinite sequence of digits ${u}_{1},{u}_{2},{u}_{3},\cdots$ in base $b$ such that the numbers ${u}_{0}{b}^{n}+{u}_{1}{b}^{n-1}+\cdots +{u}_{n-1}b+{u}_{n},$ where $n=0,1,2,\cdots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \left\{2,3,4,6\right\},$ and that no unavoidable set exists in base $b=5.$ Now, we prove that there are no unavoidable sets in base $b⩾3$ if $b-1$ is not square-free. In particular, for $b=10,$ this implies that, for any finite set of prime numbers $\left\{{p}_{1},\cdots ,{p}_{k}\right\},$ there is a nonnegative integer ${u}_{0}$ and ${u}_{1},{u}_{2},\cdots \in \left\{0,1,\cdots ,9\right\}$ such that the number ${u}_{0}{10}^{n}+{u}_{1}{10}^{n-1}+\cdots +{u}_{n}$ is not divisible by ${p}_{1},\cdots ,{p}_{k}$ for each integer $n⩾0.$

## How to cite

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Dubickas, Artūras. "Truncatable primes and unavoidable sets of divisors." Acta Mathematica Universitatis Ostraviensis 14.1 (2006): 21-25. <http://eudml.org/doc/35158>.

@article{Dubickas2006,
abstract = {We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1, u_2, u_3, \dots$ in base $b$ such that the numbers $u_0 b^n+u_1 b^\{n-1\}+\dots + u_\{n-1\} b +u_n,$ where $n=0,1,2, \dots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b \in \lbrace 2,3,4,6\rbrace ,$ and that no unavoidable set exists in base $b=5.$ Now, we prove that there are no unavoidable sets in base $b \geqslant 3$ if $b-1$ is not square-free. In particular, for $b=10,$ this implies that, for any finite set of prime numbers $\lbrace p_1, \dots , p_k\rbrace ,$ there is a nonnegative integer $u_0$ and $u_1, u_2, \dots \in \lbrace 0,1,\dots ,9\rbrace$ such that the number $u_0 10^n + u_1 10^\{n-1\}+\dots +u_\{n\}$ is not divisible by $p_1, \dots , p_k$ for each integer $n \geqslant 0.$},
author = {Dubickas, Artūras},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {Prime numbers; truncatable primes; integer expansions; square-free numbers},
language = {eng},
number = {1},
pages = {21-25},
publisher = {University of Ostrava},
title = {Truncatable primes and unavoidable sets of divisors},
url = {http://eudml.org/doc/35158},
volume = {14},
year = {2006},
}

TY - JOUR
AU - Dubickas, Artūras
TI - Truncatable primes and unavoidable sets of divisors
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2006
PB - University of Ostrava
VL - 14
IS - 1
SP - 21
EP - 25
AB - We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1, u_2, u_3, \dots$ in base $b$ such that the numbers $u_0 b^n+u_1 b^{n-1}+\dots + u_{n-1} b +u_n,$ where $n=0,1,2, \dots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b \in \lbrace 2,3,4,6\rbrace ,$ and that no unavoidable set exists in base $b=5.$ Now, we prove that there are no unavoidable sets in base $b \geqslant 3$ if $b-1$ is not square-free. In particular, for $b=10,$ this implies that, for any finite set of prime numbers $\lbrace p_1, \dots , p_k\rbrace ,$ there is a nonnegative integer $u_0$ and $u_1, u_2, \dots \in \lbrace 0,1,\dots ,9\rbrace$ such that the number $u_0 10^n + u_1 10^{n-1}+\dots +u_{n}$ is not divisible by $p_1, \dots , p_k$ for each integer $n \geqslant 0.$
LA - eng
KW - Prime numbers; truncatable primes; integer expansions; square-free numbers
UR - http://eudml.org/doc/35158
ER -

## References

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