Pythagorean Primes

Question

A Pythagorean prime number $c$ can be expressed as the sum of two squares; that is $a^2 + b^2 = c$ is a Pythagorean prime number when $c$ is prime.

What is the highest known Pythagorean prime number?

Answer 1

There is not highest Pythagorean Prime because of what gammatester said,Every prime c with c≡1(mod4) is a sum of two squares. You can infinitely find larger and larger primes of this type.

Your question must be,which is the largest Pythagorean prime number that has been ever computed.

Answer: Only Computer knows.

Answer 2

The largest known primes are tabulated here. $19249*2^{13018586}+1$ is the biggest one congruent 1 modulo 4, so it's the biggest known prime sum of two squares.

EDIT: That website now gives a bigger prime congruent 1 modulo 4: $10223*2^{31172165}+1$.