Are there any non-trivial congruencies on primes?

Question

ok... as an easy homework exercise we proved that

p = 1 (mod 4); p = 3 (mod 4)

both have infinite numbers of primes as solutions. clearly,

p = 2 (mod 4); p = 0 (mod 5)

each have 1 prime as a solution, and

p = 0 (mod 6)

has no primes as a solution.

my question is: is there any congruence that has other than 0, 1, or infinite primes as a solution set? if not, is this easily proved?

Answer 1

Dirichlet proved that any arithmetic progression a.n + b, where a and b are co-prime, contains an infinite number of primes. So if a congruence p = b (mod a) has more than one prime solution, then it has an infinity of prime solutions. No, it is not easy to prove. Even Carl Friedrich Gauss never managed to construct a proof of this which satisfied him, although he must have been sure that it was true.

Answer 2

Look up Euler's sieve then use induction, ie one prime for x=0 (mod p), no primes for x=0 (mod pq), and p = qx + 1 QED

Answer 3

If x is any real positive integer then primes can be represented as:
6x + 1 or
6x - 1

Answer 4

hola linda te saludo un argentino viajero