TITLE: Is there prime number of the form$1101001000100001000001.....$after the trivial one $k(0)=11$?
QUESTION [5 upvotes]: Let

$k(0)=11$

$k(1)=1101$

$k(2)=1101001$

$k(3)=11010010001$

$k(4)=1101001000100001$

And So on....
I've checked it up to $k(120)$, and I did't find anymore prime of such form. Are there anymore prime numbers of that form ? (I just realized that only $k(6n+5)$ could be a prime (?))

REPLY [5 votes]: The formation law is clearly
$$
n_k = 2^k n_{k-1}+1
$$
with $n_1=3$
n0 = 3;
For[i = 2, i < 50, i++, n1 = 2^i n0 + 1; 
 If[PrimeQ[n1], Print[n1, " ", IntegerString[n1, 2]]]; n0 = n1]
obtaining
n = 13 -- 1101
n = 271302750695377321080849818469209754627603342031510693802940799730825845099036699701989532948734015220469369753358523432961 -- 11010010001000010000010000001000000010000000010000000001000000000010000000000010000000000001000000000000010000000000000010000000000000001000000000000000010000000000000000010000000000000000001000000000000000000010000000000000000000010000000000000000000001000000000000000000000010000000000000000000000010000000000000000000000001000000000000000000000000010000000000000000000000000010000000000000000000000000001
If the number is considered in basis $10$ then the procedure is analogous. In this case we have $n_1 = 11$ and the recurrence equation is $n_k = 10^k n_{k-1}+1$ giving

n = 1101001000100001000001000000100000001000000001000000000100000000001000000000001000000000000100000000000001000000000000001000000000000000100000000000000001000000000000000001000000000000000000100000000000000000001000000000000000000001000000000000000000000100000000000000000000001000000000000000000000001000000000000000000000000100000000000000000000000001000000000000000000000000001000000000000000000000000000100000000000000000000000000001000000000000000000000000000001000000000000000000000000000000100000000000000000000000000000001000000000000000000000000000000001000000000000000000000000000000000100000000000000000000000000000000001000000000000000000000000000000000001 -- 1101001000100001000001000000100000001000000001000000000100000000001000000000001000000000000100000000000001000000000000001000000000000000100000000000000001000000000000000001000000000000000000100000000000000000001000000000000000000001000000000000000000000100000000000000000000001000000000000000000000001000000000000000000000000100000000000000000000000001000000000000000000000000001000000000000000000000000000100000000000000000000000000001000000000000000000000000000001000000000000000000000000000000100000000000000000000000000000001000000000000000000000000000000001000000000000000000000000000000000100000000000000000000000000000000001000000000000000000000000000000000001