Max N: <1,000,001
N Fermat 4n+1 Pythagorean prime
N v12 v22 v1 v2 p1 p2

Explanation:

I was looking for a list of integers that satisfy Fermat's theorem:
An odd prime N can be expressed as:

N = v12 + v22

with v1 and v2 integers, if and only if

N = 1 + 4n (for some integer n).

I did not find one, so I made a quick one myself.

While I was at it, I used a special case of the Brahmagupta–Fibonacci identity

(a2+b2)(c2+d2) = (ac-bd)2 + (ad+bc)2

= (ac+bd)2 + (ad-bc)2

to derive the Pythagorean primes. (You can make a right triangle with edges p1 and p2 and hypotinue N).

Legend:

N = A prime number of the form 4n+1
v12+v22 = N
p12+p22 = N2