A Sieve Method for Proving Infinite Primes of the Form n2+1

A sieve method with indication of the Chinese remainder theorem has been implemented to generate near square primes which are of the form n2 + 1, where is n is a natural number. The resulting counting function which is the number of near square primes that are less than or equal to a larger natural number, N, isπns(N) ~ N1/2 x (1-rp1/p1)(1-rp2/p2)(1-rp3/p3)(1-rp4/p4)…(1 rpy/py) + πns(py)> 2C(py) x N1/2/[ln(N)]2where py2 ~ N, rpx, x = 1, 2, 3, 4, ….., y, are the numbers of removed residue classes (0 mod px), and C(py) = (1-1/22)(1-1/42)(1-1/62)…..[1-1/(py -1)2] is the twin prime constant according to the first Hardy-Littlewood conjecture. Thus, the conjecture of Landau's fourth problem that there are infinitely many near square primes of the form n2 + 1 has been affirmed since N1/2/[ln(N)]2 goes to infinity when N is infinitely large.