

Graham's Number - grhmc
https://en.wikipedia.org/wiki/Graham%27s_number

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CurtMonash
It seems like the real character here is Graham, who I've never met, but I do
have a story about his collaborator Bruce Rothschild, who was my professor for
junior-level linear algebra.

In those days, you either gave your professors little self-addressed stamped
postcards to tell you your grade, or went to their offices to ask them. I took
the latter approach with Rothschild. "What grade were you expecting?", he
replied. I confessed that I thought I would probably get an A. "You're getting
an F", he said. "I'm giving everybody the opposite of what they expect."

~~~
pmelendez
Was he joking?

~~~
CurtMonash
Yes. Anything else would have been ... problematic.

~~~
ars
You should have told him "I expected you to say that" and let infinite
recursion reign free. :)

~~~
CurtMonash
No, that's something to do in a Hilary Putnam philosophy class. He's the one
who announces at the beginning of the term there will be a two-part final:

A. Write a question suitable for this course. B. Answer it. You will be graded
on both parts.

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ctdonath
Here's an excellent and brain melting expose of Graham's Number:
[http://waitbutwhy.com/2014/11/1000000-grahams-
number.html](http://waitbutwhy.com/2014/11/1000000-grahams-number.html)

~~~
shurcooL
Thanks for finding it and linking. That has got to be one of the most
entertaining (and educational) math articles that anyone can read/understand.

Some of my favorite parts (minor spoilers):

> Look closely at that drawing until you realize how not okay it is. Then
> let’s continue.

> Then Graham decides that for g2, he’ll just do the same thing as he did in
> g1, except instead of four arrows, there would be NO I CAN’T EVEN arrows.

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prezjordan
My brain melts every time I watch this video [1] from Numberphile on Graham's
number.

They also had another video [2] with Ron Graham himself :)

[1]:
[https://www.youtube.com/watch?v=XTeJ64KD5cg](https://www.youtube.com/watch?v=XTeJ64KD5cg)
[2]:
[https://www.youtube.com/watch?v=GuigptwlVHo](https://www.youtube.com/watch?v=GuigptwlVHo)

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torthrw
Nothing really trumps this article on big numbers.
[http://www.scottaaronson.com/writings/bignumbers.html](http://www.scottaaronson.com/writings/bignumbers.html)

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a3_nm
There was a contest to write the 512-char C program that generated the largest
number (but still terminates), assuming that all integer types are unbounded:
[http://djm.cc/bignum-results.txt](http://djm.cc/bignum-results.txt)

The method used by the winning entry is extremely interesting: return the
outputs of all possible programs up to a certain size which terminate
according to the calculus of constructions. It's quite crazy to have done this
in 512 chars.

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foklepoint
Graham is the coolest UCSD professor I have ever had a conversation with. I
remember a particular instance, at the beginning of my first quarter at UCSD.
I wasn't familiar with his work and had never heard of him. My naive 19 year
old self walked into his office hours asking some simple probability question.
I noticed a cast on his hand and inquired about it. He jokingly said he got a
cut because he was juggling ice skates. I thought he was joking. I went home
and looked him up, only to find that he was the head of the national jugglers
association at one point. Pretty awesome guy.

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simoncarter
Before clicking on the link, I thought for some reason this would be article
explaining how Paul Graham had got his own number, much like the Erdos number
[http://en.wikipedia.org/wiki/Paul_Erd%C5%91s#Erd.C5.91s_numb...](http://en.wikipedia.org/wiki/Paul_Erd%C5%91s#Erd.C5.91s_number).

~~~
nevdka
How many steps removed are you from starting a company with Paul Graham...

~~~
ikeboy
I read hacker news, so 1?

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Ronsenshi
I love this video
[http://www.youtube.com/watch?v=1N6cOC2P8fQ](http://www.youtube.com/watch?v=1N6cOC2P8fQ)
by Sean Day[9] Plott describing Graham's number very well. At least I found
that description very easy to follow. Numberphile [1] too has a great
explanation, but I still strongly recommend watching Day[9] video because it
tells a little bit of history and Sean is very expressive and good speaker so
he easily engages viewer.

[1]
[https://www.youtube.com/watch?v=XTeJ64KD5cg](https://www.youtube.com/watch?v=XTeJ64KD5cg)

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leni536
I think fast growing functions could be interesting as well. My favorite
example is the "busy beaver function"[1]. It has been shown that it grows
faster asymptotically than any computable function, where computability
roughly means that you can write an algorithm to calculate its value (without
memory or running time constrains).

Note that the Graham number can be written as G=f[64](4), where [.] means the
number of iterations of f and f(4)=3\up\up\up\up3 (as in the article). Now you
can define a function g(n) as g(n):=f[n](4), so G=g(64). g is computable since
we just described an algorithm to calculates its values. So the busy beaver
grows even faster (asymptotically) than g.

[1]
[http://en.wikipedia.org/wiki/Busy_beaver](http://en.wikipedia.org/wiki/Busy_beaver)

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excalq
Related to that is Tetration, which can be expressed with complex numbers
resulting in some beautiful fractal graphs:
[https://en.wikipedia.org/wiki/Tetration#Extensions](https://en.wikipedia.org/wiki/Tetration#Extensions)

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markrages
Mentioned in xkcd: [http://xkcd.com/207/](http://xkcd.com/207/)

~~~
thret
Now I'm horrified too.

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to3m
Interesting that they feel the need to specify which problem this number is an
upper bound for...

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noobermin
"[T]he observable universe is far too small to contain an ordinary digital
representation of Graham's number, assuming that each digit occupies one
Planck volume."

Sheesh.

