For a positive integer k, let Sk be the set of numbers n > 1 that are expressible as n = kx + 1 for some positive integer x. The set Sk is closed under multiplication. That is: If a, b Sk then ab Sk.

Definition. Suppose n Sk. If n is expressible as n = ab for some a, b Sk, then n is called k-composite. Otherwise n is called a k-prime.

For example, S4 = {5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, . . . }. The numbers 25, 45, 65, 81, . . . are 4-composites, while 5, 9, 13, 17, 21, 29, . . . are 4-primes.

1. Which n S4 are 4-primes? (Answer in terms of the standard prime factorization of n.) Show: Every n S4 is either a 4-prime or a product of some 4-primes. But "unique factorization into 4-primes" fails. To prove that, find some n = p1p2 ps and n = q1q2 qt where each pj and qk is a 4-prime, but the list (q1, . . . , qt) is not just a rearrangement of the list (p1, . . . , pr).

2. Which n S3 are 3-primes? Is there unique factorization into 3-primes?

3. Suppose a positive integer k is given, along with its standard prime factorization. Which integers n Sk are k-primes? For which k does the system Sk have unique factorization into k-primes?

Prove that your answers are correct.