Question 166
An integer $c$ is square-friendly if it has the following property: For every integer $m$, the number $m^{2}+18 m+c$ is a perfect square. (A perfect square is a number of the form $n^{2}$, where $n$ is an integer. For example, $49=7^{2}$ is a perfect square while $46$ is not a perfect square. Further, as an example, $6$ is not square-friendly because for $m=2$, we have $(2)^{2}+(18)(2)+6=46$, and $46$ is not a perfect square.)
In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following:
a. Find a square-friendly integer, and prove that it is square-friendly.
b. Prove that there cannot be two different square-friendly integers.
In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following:
a. Find a square-friendly integer, and prove that it is square-friendly.
b. Prove that there cannot be two different square-friendly integers.