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Question Number 218202 by Rojarani last updated on 01/Apr/25

Commented by Frix last updated on 01/Apr/25

$$\mathrm{I}\mathrm{think}{\mathrm{it}}^{\prime }\mathrm{s}9-4\sqrt{5}$$

Commented by Rojarani last updated on 01/Apr/25

$$Sircorrectanswer,butsolution?$$

Commented by Ghisom last updated on 06/Apr/25

$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{wrong}:9-4\sqrt{5}\approx .056$$$$\mathrm{I}\mathrm{found}$$$$-60279+246\sqrt{60043}\approx .036$$$$\mathrm{I}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{prove}\mathrm{it}\mathrm{but}\mathrm{there}\mathrm{should}\mathrm{be}\mathrm{no}$$$$\mathrm{minimum}>0$$$$$$$$\mathrm{from}a+b\sqrt{c}=0\wedge b=10-a-c\mathrm{we}\mathrm{get}$$$$a=-\frac{(c-10)\sqrt{c}}{\sqrt{c}-1}$$$$\mathrm{but}a\in \mathbb{Z}\underset{\mathrm{to}\mathrm{get}]}{\stackrel{[\mathrm{easy}}{\Rightarrow }}$$$$a=9+\lfloor \frac{9}{\sqrt{c}-1}-\sqrt{c}(\sqrt{c}+1)\rfloor$$$$b=-c+1-\lfloor \frac{9}{\sqrt{c}-1}-\sqrt{c}(\sqrt{c}+1)\rfloor$$$$\mathrm{let}f\left(c\right)=\frac{9}{\sqrt{c}-1}-\sqrt{c}(\sqrt{c}+1)$$$$a+b\sqrt{c}=(\sqrt{c}-1)(f\left(c\right)-\lfloor f\left(c\right)\rfloor )$$$$\mathrm{somebody}\mathrm{try}\mathrm{further}...$$

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