What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

• Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
• Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

• The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
• In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
• Σ(1) = 1
• Σ(4) = 13
• Σ(6) > 10¹⁰10¹⁰10¹⁰10¹⁰10¹⁰10¹⁰10¹⁰10 (10s are stacked on each other)
• Σ(17) > Graham’s Number
• Σ(27) If you can compute this function the Goldbach conjecture is false.
• Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

• YouTube - The Busy Beaver function by Mutual Information
• YouTube - Gödel’s incompleteness Theorem by Veritasium
• YouTube - Halting Problem by Computerphile
• YouTube - Graham’s Number by Numberphile
• YouTube - TREE(3) by Numberphile
• Wikipedia - Gödel’s incompleteness theorems
• Wikipedia - Halting Problem
• Wikipedia - Busy Beaver
• Wikipedia - Riemann hypothesis
• Wikipedia - Goldbach’s conjecture
• Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution