primes of the form q^2-q+1 with q=2^p-1 for prime p

[Max]

Hello!

This problem is somewhat related to Mersenne numbers.

Let p be a prime number and q = 2^p - 1 (so q is a Mersenne number but
not necessary prime). Can the number q^2 - q + 1 be prime?

Substituting the value for q, we can formulate the same problem as follows:
Can the number P(p) = 2^(2*p) - 3*2^p + 3 be prime for prime p?

What is known:

If p is not limited to primes, then there are many primes P(p).
There are also many primes P(p) if p is limited to powers of primes.
But no prime P(p) found for primes p < 150000.

Can anybody prove that P(p) can never be prime for prime p or
find prime P(p) for some prime p (in which case p must be > 150000) ?

Thanks,
Max

P.S. The origin of this problem (in russian):
http://www.nsu.ru/phorum/read.php?f=29&i=5095&t=5095