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Primes $p$ of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

it is conjectured that the number $$(2^k-1)\cdot 10^m+2^{k-1}-1$$ where $m$ is the number of decimal digits of $2^{k-1}-1$ (See also the title), is never prime when it is of the form $7s+6$. Amazingly, primes with the other residues exists although the residue $6$ occurs twice often than $1$.

Upto $k=131\ 000$, there is no counter-example, but probably a counter-example exists because the growth rate is roughly $4^k$ and $1$ out of $9$ $k's$ lead to a number of the form $7s+6$.

How large will the smallest counterexample probably be ?

Peter
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    Using Taneli Huuskonen's characterization from the linked question, it makes sense to look at all cases where $m,k$ fall into the appropriate congruence classes, but with no restriction between the sizes of $m$ and $k$. There could be local factors at some small primes that are enough to explain the difference in frequency. – Erick Wong Feb 10 '18 at 07:11
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    @Peter can be proven that 31 is the only prime of this form which can be expressed also as 2^q-1? – Enzo Creti Feb 11 '18 at 13:42
  • @EnzoCreti I have no proof that $31$ is the only one, but upto $k=10^5$ , it is the only one and I think there is no other. – Peter Feb 11 '18 at 16:19
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    It is fairly elementary to show that 31 is the only prime of that form (compared to your conjectures). If $(2^k-1)10^m + 2^{k-1}-1 = 2^l - 1$, we must have $2^k \ | \ (2^k - 1)10^m$. If $k \geq 2$ them $m < k$ so this will not be possible. – Hw Chu Feb 16 '18 at 00:41
  • @EnzoCreti Since the search gets exhausting, and the chance to find a prime "soon" is small, I decided to finish the search. According to my search, the range $[100\ 000,133\ 432]$ does not contain a counterexample. I passed the $80\ 000$ digit-mark. But due to the many candidates ($953$ in the range $[133\ 433,200\ 000]$ without a prime factor smaller than $10^7$) , I still tend to believe that there is a counter-example. – Peter Feb 16 '18 at 10:48
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    @Peter it is a bit sad that nobody but you has helped me in the search. Ok they are not Fermat primes they are not primorial primes, but perhaps it would be interesting to know if a counter-example exist. Anyway I thank you. – Enzo Creti Feb 16 '18 at 11:14
  • @Erick Wong i found five-probable primes in a row congruent to 5 mod 7. That is somewhat surprising – Enzo Creti Jun 22 '18 at 10:53

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