Prime Numbers

1.5 Exercises 51
1.7. By considering numbers of the form:
n =2 2 · 3 · 5 · ...· p − 1,
prove that there exist infinitely many primes congruent to 3 modulo 4. Find
a similar proof for primes that are congruent to 2 modulo 3. (Compare with
Exercise 5.22.)
1.8. By considering numbers of the form:
(2 · 3 · ...· p) 2 +1,
prove that there are infinitely many primes ≡ 1 (mod 4). Find a similar proof
that there are infinitely many primes that are ≡ 1 (mod 3).
1.9. Suppose a, n are natural numbers with a ≥ 2. Let N = an − 1. Show
that the order of a (mod N) in the multiplicative group Z∗ N is n, and conclude
that n|ϕ(N). Use this to show that if n is prime, there are infinitely many
primes congruent to 1 modulo n
1.10. Let S be a nonempty set of primes with sum of reciprocals S