I got the following question.
Given the following:
$4n+1=x^2+y^2$ and $ n=((a^2+a)/2)+(b^2+b)/2)$
I've been asked to express "x" and "y" in terms of "a" and "b" . But I just can't get it right, I always end up with something in terms of "x" or "y".
I got to this point and can't seem to get further on my own:
$2a^2+2b^2+2a+2b+1=x^2+y^2$
Answer
Well ... you just have to make the algebraic observation / guess
$2a^2 + 2b^2 + 2a + 2b + 1 = (a+b+1)^2 + (a-b)^2$
Another possible expression is
$( \sqrt{2} a + \frac{\sqrt{2}}{2} ) ^2 + ( \sqrt{2} b + \frac{\sqrt{2}}{2} ) ^2$
Based on the coefficients, good first guess are
$a \pm b, a \pm b \pm 1$.
We are lucky that these worked.
Otherwise, w could have tried
$\alpha a \pm \beta b$
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