decimalbinary prime pairs
I've been working on a type of prime number pairs I call "decimalbinary" primes. They are primes that consist of only 0's and 1's, like the binary system (repunits are OK). When the prime is used as a binary string, the result is a prime number, too. For example, 11 is prime, and in binary, 11 is 3, which is also prime. For obvious reasons, the number must end in 1.
Here are the pairs I found (I'm sure I missed a lot): Tell me if I made any mistakes. 11, 3 (2[sup]2[/sup]  1) 101, 5 101111, 47 1011001, 89 1100101, 101 10010101, 149 10100011, 163 10101101, 173 101001001, 337 101101111, 367 101111111, 383 110111011, 443 10100101001, 1321 11011011001, 1753 10101001010101, 10837 10101110100101, 11173 101011101010101011, 178859 1111111111111111111, 524287 (2[sup]19[/sup]  1) Probably noone knows whether there exists an infinite amount of such pairs. 
If I am interpreting correctly, here is a verbal translation of your examples:
Eleven is prime. In binary 11 is 3 which is also prime. One Hundred One is prime. In binary, 101 is 5 which is also prime. One Hundred One Thousand One Hundred Eleven is prime and so is the binary equivalent of 47. Given that the number of primes is proven infinite, a corollary would be that an infinite number of decimal/binary pairs should also exist. Note the distribution curve if you graph the examples. Thats a kind of unique pattern. Fusion 
It's interesting that both 3, (whose binary equivalent is 101), and the decimal number 101 are prime.

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