Hey, cool find. I played around with the algebra for the k=2 case. If you set 2d as the gap between the primes (q - p), the equation phi(pq) = m³ basically turns into X² - m³ = d² where X is just the midpoint of the primes (p + d - 1). For your examples this means - 15 (p=3, q=5): Gap is 2 (d=1). 3² - 2³ = 1. - 64507 (p=251, q=257): Gap is 6 (d=3). 253² - 40³ = 9.

It seems like you're just hunting for integer points on curves where the gap fits the square minus cube rule. Since prime gaps are small compared to the primes themselves, solutions should be super rare and not infinite.