Four Fours/Lemmata/Two Fours/64/Solutions/6

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

$64 = 4^{\sqrt {\map {\pi} {4!} } }$

where $\map \pi \cdot$ represents the prime-counting function.

Proof

The primes less than $4! = 24$ are:

$2, 3, 5, 7, 11, 13, 17, 19, 23$

so $\map {\pi} {4!} = 9$.

Thus:

$4^{\sqrt {\map {\pi} {4!} } } = 4^{\sqrt 9} = 64$

$\blacksquare$

Sources

• 1964: Donald E. Knuth: Representing Numbers using Only One Four (Math. Mag. Vol. 37: pp. 308 – 310)  www.jstor.org/stable/2689238