Wednesday, November 2, 2016

How big is 10^100^100?

Elliot Temple posed the following math puzzle: How many square roots does it take to get 10100100 down to a small number? (And what number is it?)

Note that abc = a(bc).

Answer: 665 square roots takes you down from 10100100 to around 4.5.

Proof:

Here, lg means the base 10 logarithm, and dot (·) means multiplication. I will use the following facts:

  • (ab)c = abc
  • xaxb = xa+b
  • x = 10lg x

Lemma:

abc = 1010c lg b + lg lg a

Example: 223 = 28 = 256 = 10103 lg 2 + lg lg 2

Proof:

 abc
=(10lg a)(10lg b)c
=(10lg a)10c lg b
=10lg a · 10c lg b
=1010lg lg a · 10c lg b
=1010c lg b + lg lg a

We can now use the lemma to show that 4.52665 = 1010665 lg 2 + lg lg 4.5. And 665 lg 2 + lg lg 4.5 ≈ 200.▢

You can also check the answer with Robert Munafo's Hypercalc, an online calculator that works with very large integers. To do this, enter 4.5^(2^665) in Hypercalc. It evaluates it to something very close to 10^10^200.

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