World Champion Largest Number - Graham's Number (instant headache!)

The Penguin Dictionary of Curious and Interesting Numbers (book) ^ | 1986 | David Wells

[ProtectOurFreedom]

The World Champion largest number, listed in the latest Guinness Book of Records, is an upper bound, derived by R. L. Graham, from a problem in a part of combinatorics called Ramsey theory.

Graham’s number cannot be expressed using the conventional notation of powers, and powers of powers. If all the material in the universe were turned into pen and ink it would not be enough to write the number down. Consequently, this special notation, devised by Donald Knuth, is necessary.

3^3 means ‘3 cubed’, as it often does in computer printouts.

3^^3 means 3^(3^3), or 3^27, which is already quite large: 3^27 = 7,625,597,484,987, but is still easily written, especially as a tower of 3 numbers: 3³³.

3^^^3 = 3^^(3^^3), however, is 3^^7,625,597,484,987 = 3^(7,625,597,484,987^7,625,597,484,987), which makes a tower of exponents 7,625,597,484,987 layers high.

3^^^^3 = 3^^^(3^^^3), of course. Even the tower of exponents is now unimaginably large in our usual notation, but Graham’s number only starts here.

Consider the number 3^^^...^^^3 in which there are 3^^^^3 arrows. A largish number!

Next construct the number 3^^^...^^^3 where the number of arrows is the previous 3^^^...^^^3 number.

An incredible, ungraspable number! Yet we are only two steps away from the original ginormous 3^^^^3. Now continue this process, making the number of arrows in 3^^^...^^^3 equal to the number at the previous step, until you are 63 steps, yes, sixty-three, steps from 3^^^^3. That is Graham’s number.

There is a twist in the tail of this true fairy story. Remember that Graham’s number is an upper bound, just like Skewes’ number. What is likely to be the actual answer to Graham’s problem? Gardner quotes the opinions of the experts in Ramsey theory, who suspect that the answer is: 6. David Wells, Dictionary of Curious and Interesting Numbers. 1986

• Other notations for writing really big numbers
• Eliezer S. Yudkowsky’s description of Graham’s Number as a Great Big Number. “This number is far larger than most people’s conception of infinity.”
• A brief description of the problem for which Graham’s Number is an upper bound. It also states that “More recently, Exoo (2003) has shown that must be at least 11 and provides experimental evidence suggesting that it is actually even larger.” But, presumably, nowhere near as large as G.
• The wikipedia article states that the lower bound was improved “to 13 by Jerome Barkley in 2008”.
• July 2020: thanks to Paul Epstein, who pointed out a mistake in Wells’ original statement about 3^^^3. It should in fact read:
  • 3^^^3 = 3^^(3^^3), however, is 3^^7,625,597,484,987 = 3^3^...^3 (7,625,597,484,987 terms), which makes a tower of exponents 7,625,597,484,987 layers high.
• While G is large, there are larger. The 18th Busy Beaver number has been proved larger.

[ProtectOurFreedom]

As long as we are discussing the Maricopa County vote tabulations here and here, I thought it would sensible to discuss IMPOSSIBLY BIG NUMBERS in the context of Dem voting crimes.

[Scrambler Bob]

(G+1)

[Scrambler Bob]

Infinity - 1

[ProtectOurFreedom]

It’s weird the author didn’t discuss that. “G+1” would appear to be the number of fraudulent manufactured “D” votes in Maricopa County. We could call it G + MC.

[Seaplaner]

3^3^...^3 + 1

[Larry Lucido]

Talking about the federal budget?

[I want the USA back]

Good. I couldn’t sleep last night worrying about how we’re going to represent the number if dollars needed to buy a loaf of bread after 4 years of biden.

[PatriotarchyQ]

Isn’t this the number of votes Trump claims he was robbed of in 2020?

[texas booster]

Donald E Knuth.

There are a lot of very smart people in this ole world, but Dr Knuth stands above them.

There may be some that are called "smarter" but in terms of applied math and science, he is way, way up there.

Yes, the same Dr Knuth that wrote the massive Art of Computer Programming way back in 1973.

[GaltAdonis]

"Oh Deep Thought - Give us the ultimate answer to life the universe and Everything!"

> 42 <

[Carl Vehse]

Compared to Rayo's Number, Graham's number can be counted on your fingers.

For details see Prof. Tony Padilla's explanation in The Daddy of Big Numbers (Rayo's Number) on YouTube.

[Fungi]

So what’s the point? Not enough material in the universe to write the number. How is it relevant to anything if it can’t be used? Get a life.

[ProtectOurFreedom]

“Claims”???

It’s a proven fact.

[ProtectOurFreedom]

“Get a life”??

I’ve proven fraud in Maricopa County and you tell me to “get a life”?

[Reily]

George Cantor territory!

[Scrambler Bob]

3^3^...^3 + 1

Yes.

What is

4^4^...^4

[Scrambler Bob]

Not enough material in the universe to write the number.
= = =

Start with those Chinese blank sheets of paper.

[Seaplaner]

Thanks. My computer just caught fire.

[Scrambler Bob]

Good Laugh.

Say, can we tweet that number to our adversaries’ computers?

Or send it in an e-mail to the DNC?

Or put it in my browser history for the FIB to find!

[Drew68]

Tree (3) is larger than Graham’s Number.