2 Positive CS-sets
The following is an extraordinary generalization attributed to the French mathematician Liouville [2], [5].
Proposition 1. For each natural n, denote by τ(n) the number of positive divisors of n. Then hτ(d) :
d|n, d ≥ 1i is a CS-set, where τ is taken over the positive divisors of n.
Proof. The proposition is clearly true for prime powers since for each prime p and each natural n, the
corresponding set of p
n−1
is