Large numbers info and calculations with programming languages.
Q: Can you calculate 3²⁷ (or 3↑↑3) by hand?
The answer is… 7625597484987, very large, right?
But it is only a 13-digit number.
What about 3⁷⁶²⁵⁵⁹⁷⁴⁸⁴⁹⁸⁷ (or 3↑↑4)?
Do not try it with Python 3, or you will get a MemoryError.
Well, it equals to 9³⁸¹²⁷⁹⁸⁷⁴²⁴⁹³·⁵ , and it is about 10³⁵⁵⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰ .
It is about 3550000000000-digit number!
Next, there is 2 ways of common representations. They are Knuth's up-arrow notation and Conway's chain.
All the arrows and chains are calculated from right to left.
For example:
x ↑ y = xʸ
x ↑↑ y = x ↑² y = ʸx = x ↑ x ↑ …(“x” will be repeated for y times)… ↑ x ↑ x
x ↑ᶻ y = x ↑↑↑…(“↑” will be repeated for z times)…↑↑↑ y
x ↑ᶻ y = x ↑ᶻ⁻¹ x ↑ᶻ⁻¹ …(“x” will be repeated for y times)… ↑ᶻ⁻¹ x ↑ᶻ⁻¹ x
1 ↑ⁿ m = 1
2 ↑ⁿ 2 = 4
m ↑ⁿ 1 = m
x → y → z = x → (x → (y - 1) → z) → (z - 1)
x → y → 1 = x → y
x → y = xʸ (if the chain length is 2)
x → y → z = x ↑ᶻ y (if the chain length is 3)
x → 1 → n = x
Let us start.
S: 4 ↑↑ 4
= 4 ↑ 4 ↑ 4 ↑ 4
= 4 ↑ 4 ↑ 4⁴
= 4 ↑ 4 ↑ 256
= 4 ↑ 4²⁵⁶
= 4 ↑ 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
= 4¹³⁴⁰⁷⁸⁰⁷⁹²⁹⁹⁴²⁵⁹⁷⁰⁹⁹⁵⁷⁴⁰²⁴⁹⁹⁸²⁰⁵⁸⁴⁶¹²⁷⁴⁷⁹³⁶⁵⁸²⁰⁵⁹²³⁹³³⁷⁷⁷²³⁵⁶¹⁴⁴³⁷²¹⁷⁶⁴⁰³⁰⁰⁷³⁵⁴⁶⁹⁷⁶⁸⁰¹⁸⁷⁴²⁹⁸¹⁶⁶⁹⁰³⁴²⁷⁶⁹⁰⁰³¹⁸⁵⁸¹⁸⁶⁴⁸⁶⁰⁵⁰⁸⁵³⁷⁵³⁸⁸²⁸¹¹⁹⁴⁶⁵⁶⁹⁹⁴⁶⁴³³⁶⁴⁹⁰⁰⁶⁰⁸⁴⁰⁹⁶
≈ 10⁸⁰⁴⁴⁶⁸⁴⁷⁵⁷⁹⁶⁵⁵⁵⁸²⁵⁹⁷⁴⁴⁴¹⁴⁹⁹⁸⁹²³⁵⁰⁷⁶⁷⁶⁴⁸⁷⁶¹⁹⁴⁹²³⁵⁵⁴³⁶⁰²⁶⁶³⁴¹³⁶⁸⁶⁶²³³⁰⁵⁸⁴¹⁸⁰⁴⁴¹²⁸¹⁸⁶⁰⁸¹¹²⁴⁵⁷⁸⁹⁰⁰¹⁴²⁰⁵⁶⁶¹⁴⁰¹⁹¹¹⁴⁹¹¹⁸⁹¹⁶³⁰⁵¹²²⁵²³²⁹⁶⁸⁷¹⁶⁷⁹⁴¹⁹⁶⁷⁸⁶⁰¹⁸⁹⁴⁰³⁶⁵⁰⁴⁶⁰
S: 2 ↑³ 6
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑↑ 2
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑↑ 4
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑ 2 ↑ 2 ↑ 2
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑ 2 ↑ 4
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑ 2⁴
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 ↑ 16
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 2¹⁶
= 2 ↑↑ 2 ↑↑ 2 ↑↑ 65536
S: 3→3→3
= 3 → (3 → (3 - 1) → 3) → (3 - 1)
= 3 → (3 → 2 → 3) → 2
= 3 → (3 → (3 → (2 - 1) → 3) → (3 - 1)) → 2
= 3 → (3 → (3 → 1 → 3) → 2) → 2
= 3 → (3 → 3 → 2) → 2
= 3 → (3 → (3 → (3 - 1) → 2) → (2 - 1)) → 2
= 3 → (3 → (3 → 2 → 2) → 1) → 2
= 3 → (3 → (3 → (3 → (2 - 1) → 2) → (2 - 1))) → 2
= 3 → (3 → (3 → (3 → 1 → 2) → 1)) → 2
= 3 → (3 → (3 → 3)) → 2
= 3 → (3 → 3³) → 2
= 3 → (3 → 27) → 2
= 3 → (3²⁷) → 2
= 3 → 7625597484987 → 2
= 3 ↑↑ 7625597484987