I am trying to find a way to describe all integer values of $x$ for which the following holds true:
$\sqrt[2]{(1/2) * x * (x - 1) + (1/4)} + (1/2)\in \mathbb{Z}$
Noting that this can be equivalently stated as:
$2*\sqrt[2]{(1/2) * x * (x - 1) + (1/4)} \equiv 1 (mod 2)$
I am aware of the fact that, via Hensel's Lemma, if I can make the left-hand side a polynomial with integer coefficients, then the solutions can easily be found. Thus, I'm trying to manipulate this congruence to make it a polynomial congruent to $0$ modulo a prime. I believed that perhaps the following was equivalent to the above expression:
$4*((1/2)*x*(x-1)+(1/4)) \equiv 1 (mod 4)$
(I.e., just squaring both sides of the congruence as well as the modulo.)
Which could then trivially be manipulated to show that:
$2*x*(x-1) \equiv 0 (mod 4)$
However, it is easy to see that this is no longer equivalent by a simple counter-example (e.g., the first statement fails for $x = 121$, whereas the second statement is true). Squaring both sides of a congruence (at least in this way) doesn't seem to make any sense.
Is there a technique I can use to remove the radical from this congruence, or am I going about this the wrong way? I would prefer hints or general techniques, rather than an explicit solution to this particular problem.
