| Title:
|
Truncatable primes and unavoidable sets of divisors (English) |
| Author:
|
Dubickas, Artūras |
| Language:
|
English |
| Journal:
|
Acta Mathematica Universitatis Ostraviensis |
| ISSN:
|
1214-8148 |
| Volume:
|
14 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
21-25 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $n \geqslant 0.$ (English) |
| Keyword:
|
Prime numbers |
| Keyword:
|
truncatable primes |
| Keyword:
|
integer expansions |
| Keyword:
|
square-free numbers |
| MSC:
|
11A41 |
| MSC:
|
11A63 |
| MSC:
|
11B50 |
| idZBL:
|
Zbl 1127.11010 |
| idMR:
|
MR2298909 |
| . |
| Date available:
|
2009-12-29T09:19:05Z |
| Last updated:
|
2013-10-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137479 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
