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?


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