Let $f(n)$ be the number of groups of order $n$ up to isomorphism. We want to prove that: $$f(a) \cdot f(b) \leq f(a \cdot b)$$ for all nonnegative integers $a$ and $b$. Our progress: If $a \cdot b \leq 2048$, then our conjecture holds. If $a + b \leq 94$, then our conjecture holds. If $\gcd(a...