This scrupulous challenge is an algebraic one, and quite interesting.
Let a “prime factor connection” be a connection from two whole numbers A and B, such that:
- A and B share a common prime factor C
- A plus C equals B
- A divided by C is not prime
For any whole number N, let the “prime connected set” of N be the set of all whole numbers that satisfy the following:
- For any number X in this set, N is part of its prime connected set
- All numbers prime connected to N are in this set
- This set is the minimum possible size given the first two requirements
Let S(N) be the size of the prime connected set of N.
Does S(N) reach infinity eventually, or is it a fast growing but strictly finite function? If it reaches infinity, can you find the smallest value at which it is infinite? Try to prove it is so. If S(N) never reaches infinity, try to prove this. Good luck!