Hi,
Prove that there exists infinetly many primes of the form:
a) 4q + 1 for some q in N U {0}
b) 4q + 3 for some q in N U {0}
Any help would be greatly appreciated.
a) consider the polynomial
Fix and suppose has a prime divisor then
By Fermat's Little Theorem: and so that is (1)
Now assume there were finitely many primes let them be and consider you can check that and so it is not divisible by a prime however this contradicts (1), since must have at least one odd prime divisor (since it's odd>1).
b) This one is easier. Suppose there were finitely many , and consider