Find all possible solutions of a square root

Question

Given the following square root: $ \sqrt{3a(148-a)} $ Find all possible natural solutions where $ a $ is an integer.

Answer 1

One can do this with a computer very easily. However, for a solution 'by hand' :-

First suppose $37$ divides $a$. Since $a\le 148$ you need only check $a\in \{0,37,74,111,148\}$. In fact all except $a=74$ give a solution.

Otherwise, and depending upon whether or not $3$ divides $a$, we have either

$a=3tu^2,148-a=tv^2$ and therefore $148=t(3u^2+v^2)$

or $a=tu^2,148-a=3tv^2$ and therefore $148=t(u^2+3v^2)$.

We are assuming $37$ does not divide $a$ and therefore a number of the form $x^2+3y^2$ is equal to either $37,74$ or $148$ (and it cannot be $74$ since $74$ is $2$ modulo $3$). The solutions are $(x,y)=(5,2),(1,7)$ and $(11,3)$.

These give solutions with $a\in \{1,27,48,100,121,147\}$.