prove that there are infinitely many prime numbers expressib

prove that there are infinitely many prime numbers expressible in the form 8n+1 where n is a positive integer.

Solution

By Fermat’s theorem, x^{p – 1} 1 (mod p) where p is an odd prime.

x⁴ –1 (mod p)

This implies that (x⁴)² (–1)² (mod p)

Or, x⁸ 1 (mod p)

The congruence x⁴ –1 (mod p) is equivalent to x⁸ 1 (mod p)

Hence, the congruence x⁸ 1 (mod p) is solvable if and only if p – 1 8k for k = 1, 2, 3,…

That is, the congruence x⁸ 1 (mod p) is solvable if and only if p 1 (mod 8)

Therefore, we can say that the congruence x⁴ –1 (mod p) is solvable if and only if p 1 (mod 8).    Proved

Prove that there are infinitely many prime numbers expressible in the form 8n + 1 where n is a positive integer.

Proof: Assume by way of contradiction that there are only finitely many of such prime numbers, say p₁,p₂,…,p_r. Consider p = 16p₁⁴p₂⁴p_r⁴ + 1 = (2p₁p₂p_r)⁴ + 1. Recall that the congruence x⁴ 1 (mod p) is solvable if and only if p 1 (mod 8). p cannot be a prime of the form 8n + 1, but p 1 (mod 8), contradiction. Thus, there must be infinitely many prime numbers of the form 8n + 1.