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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
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