Assuming finitely many: p₁,...,pₙ, then N=4p₁...pₙ-1≡3(mod4) has a prime factor ≡3(mod4) not in the list

Every product of primes ≡1(mod4) is ≡1(mod4), so we'd miss ≡3 numbers

Using Dirichlet's theorem directly

All of the above work