Four Fours/Lemmata/Two Fours/64/Solutions/4

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Puzzle: Two Fours: $64$

Using exactly $2$ instances of the number $4$, the task is to write an expression for $64$, using whatever arithmetical operations you consider necessary.


Solution

$64 = \map \phi {\map \phi {4^4} }$

where $\phi$ denotes the Euler $\phi$ function.


Proof

\(\ds \map \phi {\map \phi {4^4} }\) \(=\) \(\ds \map \phi {\map \phi {2^8} }\)
\(\ds \) \(=\) \(\ds \map \phi {2^{8 - 1} }\) Euler Phi Function of Power of 2
\(\ds \) \(=\) \(\ds 2^{8 - 2}\) Euler Phi Function of Power of 2
\(\ds \) \(=\) \(\ds 2^6\)
\(\ds \) \(=\) \(\ds 4^3\)
\(\ds \) \(=\) \(\ds 64\)

$\blacksquare$


Sources

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