Dirichlet's theorem for modulus four

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Statement

Suppose is an odd natural number. Then, there exist infinitely many prime numbers such that:

.

Facts used

  1. Congruence condition for two to be a quadratic residue
  2. Congruence condition for minus one to be a quadratic residue
  3. Nonconstant polynomial with integer coefficients and constant term of absolute value one takes infinitely many pairwise relatively prime values

Proof

We need to check two congruence classes modulo : the congruence class of and the congruence class of .

The congruence class of modulo

By fact (2), is a quadratic residue modulo if and only if . In particular, a prime can divide for some natural number if and only if .

Consider now the polynomial . For any natural number , all prime divisors of are congruent to modulo . By fact (3), there are infinitely many pairwise relatively prime values of , so we get infinitely many primes that are congruent to modulo .

Congruence class of modulo

By fact (1), is a quadratic residue modulo if and only if . In particular, a prime can divide for some natural number if and only if .

Consider now the polynomial . For any natural number , all prime divisors of are congruent to . However, itself is modulo , so must have at least one prime divisor that is modulo . By fact (3), there are infinitely many pairwise relatively prime values of , yielding infinitely many distinct primes that are modulo . In particular, all these primes are modulo .