Random challenge

Today’s challenge from the author to you: choose a random blog post from your blog and go more in-depth into it! And… oh, what’s that? The post I did on Graham’s Number? Alright!

Graham’s Number is a number in the field of googology that is formed by using Knuth’s Arrow Notation and… the number 3. Let me explain if you haven’t seen the in-depth post on the Arrow notation. Up arrow notation works as follows:

• With one arrow, it is exponentiation
• With a^ⁿb, it is a^^n-1(a^^n-1(…a^ⁿb)…))

Let’s give an example. what’s 3^3? 27. 3^^3? That’s 3^(3^3), which is 3^27, or 7.6 trillion. 3^^^3? 3^^(3^^3), which is 3^3^3^3^3^3^…3 with 7.6 trillion 3’s.

To start Graham’s Number, we need to go another step. 3^^^^3. That’s 3^^^(3^^^3), which is 3^^^(3^3^3^3^…3^3) with 7.6 trillion 3’s in the bracket. Keep in mind, even 3^^^3 was big enough. Imagine what 3^^^(3^3^3^…^3) would be.

Now, here’s the thing. Define g₁ as 3^^^^3. Define g_n to be 3^^g_n-13, or 3^^^^^^^^…^^^^^^3 with the number of up arrows being g_n-1.

Graham’s Number… is g₆₄.

How large is that?

Remember when I said what 3^^^^3 would be like? That would be like, millions and millions, maybe even quadrillions of digits, for all I know. Then take that, and calculate 3^^^^^^^^…^^^^^3 with that many ^’s. That’s only g₂, and now repeat the process 62 more times.

Yeah.