Tetration

Tetration. A bit of background on what tetration is. This notation will let you write what are, unquestionably, the biggest numbers you've ever seen in your life.

Notation for progressively larger numbers.

5 tetrated can be written as 5 to the power of 5^5^5^5 mod 10, mod 11

Consider the number 4. To make a larger number out of 4, you might add 4 to 4 and get 8. You might multiply by 4 to get 16. Or you might raise 4 to the power of itself: 4^4, or 256. But what if you took that a step further: Raise 4 to the power of 4, 4 times:

$ 4^{4^{4^{4}}} $

That's 4 tetrated, or $ {^{4} 4} $. That's a very, very, very big number:

$ 4^{4^{4^{4}}} = 4^{4^{256}} = 4^{ 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 } $

And that's just FOUR tetrated.

Imagine 10 tetrated. 20, 30, 40 tetrated. 100 tetrated. A googol tetrated.

Time for a puzzle.

What is the remainder when you divide

$ 4^{ 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096 } $

by 11?

See also: the Knuth up arrow http://mathworld.wolfram.com/PowerTower.html