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QUESTION

Find an integer $n$ that can be written as a sum of two squares in
two essentially different ways; i.e.$n=u^2+v^2=w^2+x^2$ where
$w\neq\pm u,\pm v$.

[Hint: recall $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$. What
happens when we exchange $c$ and $d$?]




ANSWER

$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$, and
$(a^2+b^2)(d^2+c^2)=(ad+bc)^2+(ac-bd)^2$.

These two expressions could turn out to be different, e.g.
$5=2^2+1^2$, and $17=4^2+1^2$. Thus, as above, we have
$85=5.17=(2^2+1^2)(4^2+1^2)=(8+1)^2+(2-4)^2=9^2+2^2$, and
$85=5.17=(2^2+1^2)(1^2+4^2)=(2+4)^2)(8-1)^2=6^2+7^2$.

[Plenty of other examples are available- so your answer may well
be different.]




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