# Truncatable primes and unavoidable sets of divisors

Acta Mathematica Universitatis Ostraviensis (2006)

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

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topDubickas, 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 -

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