# Probable primes of a particular form

Concatenating two consecutive Mersenne numbers in base 10, I found these primes/probable primes:

$$(2^{215}-1)\cdot10^{65}+2^{214}-1$$ PROVEN PRIME

$$(2^{69660}-1)\cdot10^{20970}+2^{69659}-1$$ probable prime

$$(2^{92020}-1)\cdot10^{27701}+2^{92019}-1$$ probable prime

$$(2^{541456}-1)\cdot10^{162995}+2^{541455}-1$$ probable prime

These are the only known primes/probable primes of this form with exponent multiple of $$43$$. Infact $$215, 69660, 92020, 541456$$ are multiples of $$43$$. What I noticed is that: $$\frac{215}{41}$$, $$\frac{69660}{41}$$, $$\frac{92020}{41}$$, $$\frac{541456}{41}$$ have all a decimal periodic expansion of $$\overline{24390}$$, where $$24390=29^3+1$$. I wonder if that is only coincidence or if there is some explanation. These numbers have the form $$(2^k-1)*10^d+2^{k-1}-1=P(k)$$, where d is the number of decimal digits of $$2^{k-1}-1$$. So I wonder if for k multiple of $$43$$ and P(k) prime,$$\frac{k}{41}$$ gives always a periodic decimal expansion of $$\overline{24390}$$. In other words when P(k) is prime and k is a multiple of 43, then is it always true that: k is of the form $$41s+r$$ where r is an integer in the set $$(1,10,16,18,37)$$? Link: Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?