__Theorem__ (Green-Tao) Let A be any subset of the prime numbers of positive relative upper density, thus

where denotes the number of primes less than or equal to N. Then A contains arithmetic progressions of length K for all K.

__Specialization :__ Consider the set of primes p = 1 (mod 4)

__Theorem :__ (classical) Every prime p = 1 (mod 4) can be written as the sum of two squares.

__Quadratic corollary :__ There are arbitrarily long arithmetic progressions consisting of numbers which are the sum of two squares.

__Special case :__ k = 4.

__Theorem__ (Heath-Brown) Let S be the set of sums of two squares. There are infinitely many 4-term arithmetic progressions in S.

__Proof :__ The numbers (n-1)^{2}+(n-8)^{2}, (n-7)^{2}+(n+4)^{2}, (n+7)^{2}+(n-4)^{2} and (n+1)^{2}+(n+8)^{2} form a 4-term arithmetic progression for any strictly positive integer.

__Remark :__ n = 1 : 7^{2}, 6^{2}+5^{2}, 8^{2}+3^{2}, 2^{2}+9^{2} (Degenerated cases)

n = 4 : 3^{2}+4^{2}, 3^{2}+8^{2}, 11^{2}, 5^{2}+12^{2}

n = 7 : 6^{2}+1^{2}, 11^{2}, 14^{2}+3^{2}, 8^{2}+15^{2}

n = 8 : 7^{2}, 1^{2}+12^{2}, 15^{2}+4^{2}, 9^{2}+17^{2}