I recently found myself asking if the following (diophantine) expression ever evaluates to a square number:
$$5+12n$$
I was surprised both to be unable to stumble across an integer value for $n$ that results in a square number, and then surprised not see an obvious, eloquent, proof showing why this expression can never be square.
I would be interested in any pointers to work on the question of if/when equations of the form:
$$ A + Bn = c^2 $$
have a (non-trivial) solution. I'm assuming the convention that capital variables are constants, while non-capitals are free variables.
Look modulo $B$: $$ A\equiv c^2\text{ (mod }B\text{)} $$ I don't know whether or not you're familiar with Quadratic Reciprocity, but there is plenty of information on that Wikipedia article about conditions for $A$ to be a square modulo $B$ if $B$ is an odd prime. If there is no solution modulo $B$, then there is no solution in general.
In your specific case, $5$ is not a square modulo $12$ (you can easily check that squaring the numbers $1$ to $11$, and then finding their remainder when divided by $12$ will never give you $5$), so there is no solution.