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Sketching an IMVHO promising idea. There is a gap in my estimates, but may be it can filled. Let $m=\lfloor b\sqrt7\rfloor$, $\delta_1=b\sqrt7-m$, $n=\lfloor a\sqrt2\rfloor$, $\delta_2=a\sqrt2-n$. The first thing we need is a lower bound for the round off errors $\delta_1$ and $\delta_2$. $$ ...



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