Here :

https://math.stackexchange.com/questions/2635516/a-conjecture-about-numbers-n-of-the-form-10m2k%e2%88%9212k-1%e2%88%921-where-m/2636195#comment5529838_2636195

Enzo Creti asks for a prime number formed by concatenating the Mersenne-numbers 2^n-1 and 2^(n-1)-1 , for example 40952047. For all residues, modulo 7, he found primes except of the residue 6. This is somewhat surprising because the residue 1 occurs only with half frequency.

> Is there any hidden structure forcing a non-trivial factor in the case of residue 6, or was it just "bad luck", that no prime was found despite an enormous search range ?

I invite anyone to join in the search of a prime. I posted the necessary details on github, which is linked in the linked question.