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# Trying to prove pells equation doesn't work with square numbers watch

1. Hello, I am trying to prove that for pells equation "" when n is a perfect square the equation has no solutions. Given that all x, y and n are Natural numbers not including zero.
So far I have got this:

(difference of two squares)

So from here am I right in saying that the left hand side is going to be 2 or higher and the right surely must be 1 or less? I feel like I'm missing a final step or possibly need to word my statement better, any ideas?
2. (Original post by iAustinMark)
Hello, I am trying to prove that for pells equation "" when n is a perfect square the equation has no solutions. Given that all x, y and n are Natural numbers not including zero.
So far I have got this:

(difference of two squares)
We have two cases: or .
3. (Original post by iAustinMark)
Hello, I am trying to prove that for pells equation "" when n is a perfect square the equation has no solutions. Given that all x, y and n are Natural numbers not including zero.
So far I have got this:

(difference of two squares)

So from here am I right in saying that the left hand side is going to be 2 or higher and the right surely must be 1 or less? I feel like I'm missing a final step or possibly need to word my statement better, any ideas?
You are home and dry on the second to last line: you have the product of two integers equal to one. What are the possible solutions?
4. (Original post by Gregorius)
You are home and dry on the second to last line: you have the product of two integers equal to one. What are the possible solutions?
So (x + my) and (x - my) would have to both equal plus or minus 1? That not being possible if all x,y and m are integers, is that proof enough in itself?
5. (Original post by iAustinMark)
So (x + my) and (x - my) would have to both equal plus or minus 1? That not being possible if all x,y and m are integers, is that proof enough in itself?
You now have two sets of equations (one each for +1 , -1) in two unknowns. Solve them. x+my=1 and x-my=1 implies x=? and y=?
6. (Original post by Gregorius)
You now have two sets of equations (one each for +1 , -1) in two unknowns. Solve them. x+my=1 and x-my=1 implies x=? and y=?
I suppose it would imply x=1 and y=0 but that's not possible because y can't be 0
7. (Original post by iAustinMark)
I suppose it would imply x=1 and y=0 but that's not possible because y can't be 0
Yes.

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