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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.

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$m$ is $\{2,3,5\}$-smooth iff $5m$ is, hence $5m$ and $5m\pm1$ are consecutive $\{2,3,5\}$-smooth numbers - exactly what Størmer's theorem is about.

The case of consecutive $\{2,3,5\}$-smooth number happens to be treated as an example for Lehmer's method in Wikipedia

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    (embarrassment...)2012-10-08