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        riddles >> medium >> Squares: sums and differences
        
(Message started by: Christine on Mar 11^th, 2013, 1:23pm)

Title: Squares: sums and differences
Post by Christine on Mar 11^th, 2013, 1:23pm

My question is: how do you find sums and differences that are both squares?

x + y is a square
x - y is a square

Title: Re: Squares: sums and differences
Post by towr on Mar 11^th, 2013, 1:44pm

(u+v)^2-u^2 = 2uv+v^2 = 2y
x = u^2+y
So basically we can take any u paired with an even v.

Title: Re: Squares: sums and differences
Post by pex on Mar 11^th, 2013, 1:46pm

Or, equivalent but perhaps simpler: take any two squares that are either both even or both odd, say a² and b², and let x = (a² + b²)/2 and y = (a² - b²)/2.

Title: Re: Squares: sums and differences
Post by Immanuel_Bonfils on Mar 11^th, 2013, 5:01pm

No restriction to x and y (unless being real, y guess), so

parity doesn't matter, even b>a -> (y <0) would be OK

Title: Re: Squares: sums and differences
Post by Grimbal on Mar 12^th, 2013, 4:00pm

It wasn't stated, but usually these problems ask for integer solutions. If you are working with reals, every number >=0 is a square.

A simple set of solutions is x=a² for some a and y=0. It doesn't feel like a proper solution though.

On the other side, pex's method gives all integer solutions.

Title: Re: Squares: sums and differences
Post by Christine on Mar 19^th, 2013, 10:34am

Thank you all for the feedback.

Sorry I did not state that
> I was looking for integer solutions,
> for producing interesting results (e.g. squares of prime numbers)

I'll make sure that next time I'll state clearly my questions.