Power tower (tetration): a↑↑b = a^(a^(a^...)) with b copies. 2↑↑4 = 2^(2^(2^2)) = 2^(2^4) = 2^16 = 65536. 3↑↑3 = 3^(3^3) = 3^27 = 7,625,597,484,987. Growth is incomprehensibly fast: 3↑↑4 has over 3.6 trillion digits.
How to Use
Enter a and b:
- a↑↑b: The power tower
- Last digits: Modular result
- Exact: For small values
Computing Last Digits
Euler's theorem: a^φ(m)≡1 (mod m). Recursively: a^x mod m depends on x mod φ(m), then x mod φ(φ(m)), etc. Since φ iterated reaches 1, power towers modulo m always stabilize.
Hyperoperations
- a+b: addition (level 1)
- a×b: multiplication (level 2)
- a^b: exponentiation (level 3)
- a↑↑b: tetration (level 4)
- a↑↑↑b: pentation (level 5)
Step-by-Step Instructions
- 1Enter base a.
- 2Enter height b.
- 3View result.
- 4See last digits.
- 5Compare values.