Classification of quadruples with gcd(ar, bs) = 2; specific case (a, b, r, s) = (3, 5, 2, 2)

Determine whether there exist positive integer quadruples (a, b, r, s) with gcd(ar, bs) = 2, beyond the cases (1, 1, 2, 2) and the families (a, 1, r, 2) with odd a > 1 and even r, and (1, b, 2, s) with odd b > 1 and even s, such that every positive rational number can be written as ((φ(m^r))^a)/((φ(n^s))^b) for some m, n ∈ ℕ; in particular, ascertain whether every positive rational number can be written as ((φ(m^2))^3)/((φ(n^2))^5) for some m, n ∈ ℕ.

Background

The paper establishes several exclusion results: if gcd(ar, bs) = d > 1 and d ∤ (a − b), or if d > 2 and d | (a − b), then ((φ(m^r))^a)/((φ(n^s))^b) cannot represent all positive rationals for any choice of m, n ∈ ℕ. Complementing these, Theorem 1 and Fact 5 provide positive families with gcd(ar, bs) = 2, namely (a, 1, r, 2) and (1, b, 2, s) for odd a, b > 1 and even r, s.

Despite these advances, the authors explicitly note that it remains unknown whether additional quadruples with gcd(ar, bs) = 2 exist. They highlight a concrete test case: whether the pair of exponents (a, b, r, s) = (3, 5, 2, 2) suffices to represent all positive rationals via ((φ(m^r))^a)/((φ(n^s))^b).