I am trying to prove, Product of primes of the form $(4k-1)$ can't be sum of 2 squares. My approach is-
Let the product is $M=m_1m_2...m_n$ where $ m_1, m_2, ...m_n$ are primes.
Assume, $M$ can be written as sum of 2 squares. Then,
$M= x^2 + y ^2 \implies m_1m_2...m_n = (x+iy)(x-iy)$
But I am stuck at this stage, there must be something related to the property of Gaussian Integer.
How can it be proved?