Let $a,b,c\in\mathbb{Z}^+$ and $a Example: Suppose $n=480$, then the two factors of $n$ such that their difference is minimum are $a=20$ and $b=24$.
Let $c=3$, then the factors of $nc^2=4320$ such that their difference is minimum are $ac=60$ and $bc=72$. However, if $c=6$, then the factors of $nc^2=17280$ such that their difference is minimum are $128$ and $135$ and not $ac=120$ and $bc=144$. The case is trivial if $n$ is a square number but is there a general condition?
A problem concerning natural numbers broken up into two factors such that their difference is minimum.
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discrete-optimization
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0There are some serious wording problems but your example makes your intent clear. The first two sentences are written backwards: this is a definition of $a,b$ given $n$, not a definition of $n$. It is not the quantity $bc-ac$ that is minimized: rather you are asking when $nc^2$ has no divisor between $ac$ and $c\sqrt{n}$. – 2017-02-08
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0@ErickWong Thank you for pointing errors in the format. Formatted as indicated. – 2017-02-08