Problem 8. In this problem, you will show that there are infinitely many prime numbers congruent to -1 mod 3.
(a) List at least seven prime numbers that are congruent to -1 mod 3.
(b) Show that if N is a positive integer whose prime factors are all congruent to 1 mod 3, then N is congruent to 1 mod 3.
(c) Suppose for a contradiction that there are only finitely many prime numbers congruent to -1 mod 3. Label them p1, ..., pk. Consider the number
N = 3p1 ··· pk - 1.
How do we know that N is a positive integer? (We know there are some primes congruent to -1 mod 3, right?)
Prove that some prime factor q of N is congruent to -1 mod 3. (Since N is not a multiple of 3, none of its prime factors are; these prime factors are therefore congruent to either 1 or -1 mod 3. What goes wrong if all of the prime factors of N are congruent to 1 mod 3?)
(d) Show that the prime factor q of N from part (c) is not in the list p1, ..., pk. Explain how this leads to a contradiction, and therefore to a proof that there are infinitely many prime numbers congruent to -1 mod 3.