Knuth's up-arrow notation

Knuth's up-arrow notation was invented by Donald Knuth in 1976, and is used to compute really big integers. Up-arrow notation can be thought of as a way to write iterated exponentiation.

Examples

Single arrow

$x\uparrow y=x^{y}$

$4\uparrow 3=4^{3}=64$

$7\uparrow 4=7^{4}=2401$

$3\uparrow -2=3^{-2}={\frac {1}{9}}$

Double arrow

$x\uparrow \uparrow y=\underbrace {x^{x^{.^{.^{x}}}}} _{y{\text{ times}}}$

$3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987$

$5\uparrow \uparrow 3=5^{5^{5}}=5^{3125}\approx 1.911013\times 10^{2184}({\text{2185 digits long}})$

$7\uparrow \uparrow 4=7^{7^{7^{7}}}=7^{7^{823543}}\approx 7^{3.759823\times 10^{695974}}\approx 10^{3.177419\times 10^{695974}}(~3.177419\times 10^{695974}{\text{ digits long}})$