The first 8 primes of the form 166...661 have lengths 5, 13, 17, 19, 37, 53, 73, and 101. It is always the case if a number of this form is prime then its length also must be prime?
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2Note that $$n_L=\frac{5\cdot10^{L-1}-17}3$$ – Peter Foreman Dec 04 '19 at 16:44
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Have not found a probable prime checking up to 4000 digits. It may be the case that these are the only prime values of the sequence? – Goldbug Dec 05 '19 at 13:57
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If $n$ is even, the number is divisible by $11$.
If $n$ is odd and divisible by $3$, the number is divisible by $7$.
That leaves $33$ possible $n$, of which $24$ are prime.
If you picked eight of the 33 at random, you would have $5.3\%$ chance of getting all prime. Can you get some more data ?
Empy2
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division by 13 starts remainder 10 before alternating 2,6 taken first 3 digits at a time. not sure what happens at the far end. – Dec 05 '19 at 03:17
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According to The On-Line Encyclopedia of Integer Sequences,
the next (probable) prime of that form has $6233$ digits,
and $6233=23×271$.
J. W. Tanner
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the next primes after that have $24029$ (prime), $40223 (19\times29\times73)$, and $66395 (5\times7^2\times271)$ digits – J. W. Tanner Dec 05 '19 at 18:20
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1If these probable primes are correct then this information answers the question... – Goldbug Dec 05 '19 at 20:38