Find all integral solutions of the equation .
My attempt: The equation can be rewritten as .
Can someone please tell me how to proceed?
I hate to be writing the very first comment in this thread without any actual help within, but it has been resting for a while now...
I noticed that
May be rewritten as (quite unusual form)
I don't have a clue whether there can be a slightest use of this though, sorry...
Let . Check that
Then , so it follows from (1) that is always too small to be a square root for
Next, , and the only integers for which this is not positive are and . For all other integers, is positive, and it follows from (3) that is too large to be a square root of
Thus, unless or , the only possible candidate for an integral square root of is , and it follows from (2) that this solution will only work if
Therefore the only values of that might give solutions to the problem are and . If then , which is not a square. But the other three values of give the solutions to the problem, namely .
The strategy is quite simple really. We want to find an integer that is a square root for . My claim is that the best candidate for this square root is . Equation (1) shows that is definitely too small to be a square root for Similarly, equation (3) shows that is too large to be a square root for except possibly when or So, for all other values of , the square root must be bigger than and smaller than , which leaves as the only remaining possibility. But equation (2) shows that for to be the square root, it is necessary that , and that only happens when