x^4+1 is prime for {2, 17, 257, 1297, 65537, 160001..} (A037896 in Sloane - see also A077497 in Sloane.) Now 160001 is 2^8*5^4+1, equivalently 2*4*10^4+1, and the same series A037896 includes both 40960001 and 655360001.

Call such prime numbers, of the form 2^(4k)*10^(4n)+1, where either or both of (4k+1) and (4n+1) are also prime, Nickki primes.

There are then 3 kinds of such primes:
(4k+1) prime, (4n+1) composite: 1st kind.
(4k+1) composite, (4n+1) prime: 2nd kind
(4k+1) prime, (4n+1) prime: 3rd kind

1. It is not hard to find Nickki primes of the 3rd kind for qualifying k up to 7, but are there any more such primes for, say, k not exceeding 25?

2. {k=5, n=1} is an example of a Nickki prime of the 2nd kind. Are there any others for, say, k not exceeding 25?

3. [harder] Is there a Nickki prime of either the 1st or 3rd kind for every qualifying k not exceeding 25?