In the context of Hurwitz groups and manifolds, one comes by what wikipedia defines as a "remarkable" fact, that $1-\dfrac1 a -\dfrac 1 b - \dfrac1 c > 0$ has a minimal value of $1/42$ if $a,b,c \in \mathbb{Z}$ and $a<b<c$ (I'm not sure this last part is needed, but let's consider it anyway).
I was wondering how one would go about solving a problem like this. I am totally at loss when I see problems "over the integers": I never actually learned any techniques to approach such problems. So the question is: is there a suitable general method to minimize expressions over the integers? I of course feel that there is not standard way, but I'm interested in some examples as to how to proceed, maybe taking this expression as a base for the examples.
For this problem you just need some intelligent case analysis. Increasing the value of $a, b$ or $c$ makes the sum of reciprocals smaller and thus increases the expression on the right hand side. So you need the smallest values you can get.
$a=1$ leaves the left hand side negative, and the best we could do with $a\ge 3$ would be to set $a=3, b=4, c=5$ giving us $\frac {13}{60}$. So we need to test $a=2$ to see if we can do better than this.
The expression then becomes $\frac12-\frac1b-\frac1c$ with $2\lt b \lt c$
If $b\ge 4$ the best we can do is $b=4, c=5$, giving us a value of $\frac 1{20}$.
So we are left testing $b=3$, when our expression becomes $\frac16-\frac1c$. The lowest value of $c$ for which this is positive is $c=7$ and this gives the minimum value $\frac1{42}$.