A number of my friends at school came to me with the following problem which I was unable to solve. For any integer $x$, there exists $y$ so that: $$ x^2 + y^2 \equiv 1 \mod x + y$$ I understand I am trying to solve for two integers $y$ and $q$ so that: $$x^2 + y^2 = r(x+y) +1$$ However, I was unable to proceed much beyond this. Experimentally, I was able to compute $y$ and $q$ for a few values of $x$ but I wasn't able to identify and patterns in these values.
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By $x^2+y^2\equiv -2xy\pmod{x+y}$, we are really asking for $y$ such that $x+y \mid 2xy+1$. But $x\equiv -y\pmod{x+y}$, so this is equivalent to finding $y$ such that $$x+y \mid 2x^2-1.$$
The existence of such a $y$ and its possible values should be clear now: for any divisor $d$ of $2x^2-1$, $y = d-x$ works. In particular, taking $d=1$ and $d=2x^2-1$ show that $y = 1-x$ and $y = 2x^2-x-1$ both work everytime.
Ian Mateus
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