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Prove or disprove that all numbers $717, 71717, 7171717,\dots$ are composite.

This is related to this question.

$\begin{array}\\ 717 &= \text{div by 3}\\ \color{blue}{71717} &= 29\cdot 2473\\ 7171717 &= \text{div by 7}\\ 717171717 &= \text{div by 3}\\ \color{blue}{71717171717} &= 857\cdot83683981\\ 7171717171717 &= \text{div by 7}\\ 717171717171717 &= \text{div by 3}\\ \color{blue}{71717171717171717} &= 11\cdot239\cdot2011\cdot13565021843\\ \end{array}$

Q: Is the subsequence in blue always composite?

Note 1: The numbers in blue are

$$F_n = \frac{71\times10^{6n-1}-17}{99}$$

and $F_n$ is composite for $n<5000$ (user Uncountable).

Also, $F_8=71717171717171717171717171717171717171717171717 =35972094619010351911⋅1993689065836910882277192947$

Note 2: For similar $737,73737,7373737,\dots$

$$P_n = \frac{73\times10^{6n-3}-37}{99}$$

$P_3 = 737373737373737$ is prime and $P_n$ is prime for $n = 3,7,95,422,2390$ for $n<5250$ (user Uncountable).

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    $\begingroup$ I've revised your question to be analogous to the link you provided. $\endgroup$ – Tito Piezas III Apr 15 '15 at 3:23
  • $\begingroup$ For Note 2, we have, $$P_n = \frac{73\cdot10^{6n-3}-37}{99}$$ and for $n<23$ is prime for $n=3,7$. Question: Is it necessarily the case that for $P_n$ (or $F_n$) to be prime, then $n$ must be prime? (I assume it is just coincidence, but I thought I should ask.) $\endgroup$ – Tito Piezas III Apr 15 '15 at 4:40
  • $\begingroup$ Is the subsequence in brown always composite ? - No. $\endgroup$ – Lucian Apr 15 '15 at 5:29
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    $\begingroup$ A related question. $\endgroup$ – Lucian Apr 15 '15 at 9:49
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    $\begingroup$ Also, if $a_0=7$ and $a_{n+1}=100~a_n+17$, then there are no primes for $n\le5,000$. $\endgroup$ – Lucian Apr 15 '15 at 9:53

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