I came across a problem recently which can be reduced to finding numbers $m$ such that $m$ and either $5m+1$ or $5m-1$ are $\{2,3,5\}$-smooth, i.e., of the form $2^a3^b5^c$ for nonnegative integers $a,b,c.$ (Of course this is the same as checking if $m(5m\pm1)$ is $\{2,3,5\}$-smooth.)
I checked up to $10^{100}$ and found only small cases: $ m\in\{1, 2, 3, 5, 16\} $
How can I prove that this list is complete? If I was looking at $m\pm1$ I could use Størmer's theorem but that's not available here.