Consider $x^2-x-a=0$, I want to find value of a for which the roots of the equation are integers $ ∀$ $ a∈ℕ $ and $a≤100$.
So the discriminant should be a perfect square.
So $1+4a=k^2$ Now I have to find all possible values of $a$.
I proceeded like this:
$4a=k^2 -1=(k+1)(k-1)$
Now When I tried to solve by breaking $4a$ into $4 · a$ and $2 · 2a$ and $4a · 1$. I could find only $2$ solutions. However there are more than that like $12$, $20$, etc.
I found the values by putting value of a and found difference between consecutive terms to be in A.P. but still couldn't figure out the general form.
How do I find them without putting values of $a$?
Hint:
Notice if we ensure $k\ge 2$ is odd, then $$a=\frac{(k-1)(k+1)}{4} $$ will be a natural number, as we get atleast one factor of $2$ each from $k-1$ and $k+1$.