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This question is O205 from the Mathematical Reflections. I do not own any copyrights to this question.

Find all $n$ such that each number containing $n$ $1$'s and one $3$ is prime.

For example, when $n=2$ we find that $113, 131$ and $311$ are prime.

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    Clearly $n$ is not a multiple of $3$.2012-03-01
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    There is a test for the divisibility by 7 (http://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_by_7, second §) which rules out any $n \geq 6$.2012-03-01
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    @D.Thomine Could you elaborate?2012-03-01
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    @D. Thomine That seems to rule out only those $n$ greater than $4$ which are not congruent to $5$ mod $6$.2012-03-01
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    @ Chris. You are right. I found my mistake.2012-03-01
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    And 17 rules out $n \geq 9, n \not= 15 \pmod{16}$, 19 rules out $n \geq 11, n \not= 17 \pmod{18}$ and 23 rules out $n \geq 17, n \not= 21 \pmod{22}$ from the limited experimental evidence I have (I could be wrong though)2012-03-01

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