# Fourier base - not as you know it!

Number Theory Level pending

A natural number $$n$$ is called foury if there exsits a natural number $$b$$ such that in base $$b$$ all the digits of $$n$$ are fours, e.g. $$4$$ is foury and so is $$624$$ as $$624_{10}=4444_{5}$$. Clearly, if $$n$$ is foury then $$n\equiv0\pmod 4$$. There exists a smallest foury number $$N$$ such that for all $$n\geq N$$ if $$n\equiv0\pmod 4$$ then $$n$$ is foury.

A natural number $$n$$ is called unfoury if in any natural base $$b$$, none of the digits of $$n$$ is four. E.g. $$1$$ is unfoury. There exists a largest unfoury number $$M$$.

Find $$N$$ and $$M$$. Your answer will be of the form $$N.M$$ (with a decimal point between them). E.g. if you think that $$N=4$$ and $$M=1$$ then you should enter $$4.1$$.

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