Can a dozenal (duodecimal, base 12) number {111...111}21{111...111} (with the same number of 1s in the two brackets) (start with 21, 1211, 112111, 11121111, ...) be prime? I cannot find such prime with <=1000 (decimal 1728) digits, but find neither covering set nor algebra factors. (I have proved that dozenal (duodecimal, base 12) numbers 1{555...555}1 (start with 151, 1551, 15551, 155551, ...) cannot be primes, because of covering sets and algebra factors)

Also, can a dozenal (duodecimal, base 12) 414141...4141411 (start with 411, 41411, 4141411, 414141411, ...) be prime? I want to find the dozenal (duodecimal, base 12) analog of

A086766, and I proved that there are no primes for n = 10 (decimal 12) and n = 33 (decimal 39), because of algebra factors, thus the conjecture in

A086766 is not true in dozenal (duodecimal, base 12).

Besides, I want to find the dozenal (duodecimal, base 12) analog of many other OEIS sequences, such as

A088782,

A069568,

A200065,

A272232,

A089776 (n%12 = 1, 5, 7, 11 instead of n%10 = 1, 3, 7, 9),

A267720,

A244424,

A262300,

A046035,

A047777,

A060421,

A064118.

For

A069568 case, I found the prime (12^1676*298-1)/11 for n = 23 (decimal 27) and proved that there are no primes for n = 34, 89, and 99 (decimal 40, 105, and 117), because of covering sets and algebra factors, but for the

A089776 case, I cannot find prime for n = 65 (decimal 77) and n = EE (decimal 143).