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Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$

There is no such finite set of pairs $(a_k,b_k)\in\mathbb{Q}_{\geq 0}^2$. Indeed, because the problem concerns rational representations, we can assume without loss of generality that $(a_k,b_k)\in\...
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