Exercise 13 establishes an interesting relation between the two multisets Spec(α) and Spec¡ α/(α − 1)  , when α is any irrational number > 1, because 1/α + (α − 1)/α = 1. Find (and prove) an...

Exercise 13 establishes an interesting relation between the two multisets Spec(α) and Spec¡ α/(α − 1)  , when α is any irrational number > 1, because 1/α + (α − 1)/α = 1. Find (and prove) an interesting relation between the two multisets Spec(α) and Spec¡ α/(α + 1) ¢ , when α is any positive real number

Exercise 13:

Let α and β be positive real numbers. Prove that Spec(α) and Spec(β) partition the positive integers if and only if α and β are irrational and