Let's say I have two natural numbers $n$ and $m$, both of which are exceeding large, and I want to determine which is smaller. Thankfully, I happen to know both of their prime factorizations, but they're composed of many terms to high powers, so it's infeasible for me to just multiply it out and determine which is smaller that way. Hence my question is: Is there any way to compare the magnitude of two numbers, knowing only their prime factors?
As a perhaps somewhat artificial example, take: $$n=67\times24977\times8598615939$$ $$m=2^5\times3^2\times5^5\times7^4\times13\times23^3\times421$$ Is there anyway to know $m>n$ from just those factorizarions apart from multiplying them out and comparing digits?