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Natural numbers in base $10$ have a finite notation. But there are other bases of the form $\frac{a+\sqrt b}{2}$, where all natural numbers can be written in a finite notation. In base $\frac{5+\sqrt 17}{2}$ for example the decimal number $11$ would be written as $21.4$.

What is the condition for possible values $(a,b)$ so that natural numbers can be written in a finite notation ?

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