Find all integral solutions of the equation .

My attempt: The equation can be rewritten as .

Can someone please tell me how to proceed?

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- Jan 15th 2011, 12:54 AMalexmahoneDiophantine equation
Find all integral solutions of the equation .

My attempt: The equation can be rewritten as .

Can someone please tell me how to proceed? - Jan 15th 2011, 12:00 PMPranas
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... - Jan 15th 2011, 04:59 PMorange gold
Re-wrote it as:

I don't know where I'm going with this, but maybe that helped :D - Jan 15th 2011, 07:31 PMtonio
- Jan 16th 2011, 01:32 AMOpalg
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 . - Jan 16th 2011, 04:26 AMroninpro
- Jan 16th 2011, 01:41 PMOpalg
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

- Jan 16th 2011, 02:06 PMchiph588@