The minimum number of digits after the floating-point, which uniquely identify every irrational square root

Question

Let the following:

• $B:$ a natural number larger than $1$
• $S:$ a set of irrational numbers in the range $(0,1)$ represented in base $B$
• $L:$ the minimal prefix length which uniquely identifies every element in $S$

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With $S=\{r|r=\sqrt[2]{n}-\lfloor\sqrt[2]{n}\rfloor,r\neq0,n\in\mathbb{N}\}$, is it possible to express $L$ as a function of $B$?

An answer for specific bases (binary and decimal in particular) would also be highly appreciated.

Answer 1

Note that $\sqrt {n+1}-\sqrt n = \cfrac 1{\sqrt {n+1}+\sqrt n}$ and can therefore be made arbitrarily small so that however many digits $L$ you choose, you can find square roots so close together that $L$ digits will not distinguish them.