Graham’s number

In the 2nd or 3rd grade of my catholic primary school our teacher spoke of stuff like ‘God’
and ‘Heaven’.
“People live in Heaven after their death,” she told us. “All eternity.”
Nobody knew what eternity was, so she had to explain.
“Eternity,” she said, “never ends. “The years go on forever.”
Apparently she saw her pupils struggling with this concept and that made her tell us a

“Somewhere far away there is an enormous mountain that is made of diamond which is, as I told you earlier, the hardest material there is. Now one time in a million years, a little bird passes by, he briefly sharpens his beak on this mountain and flies away. Not to return for another million years to sharpen again.
Of course the mountain wears out a tiny bit by this sharpening and gradually gets smaller.
And smaller.
After many, many, many portions of million years the mountain is entirely gone.
And then eternity is not passed yet.”
That awed us to put it mildly.

Decades after, I read about scientific research into the future of our universe.
More precise: the research into how long life could exist in a ever-deteriorating space-time continuum, increasingly exhausted and energy depleted.
Life (here defined as an information collecting and processing entity), would come to an end when the universe was about 10 to the hundredth years old. 10100 year!

I informed a friend about this remarkable conclusion, but he didn’t grasp.
“You mean, the age of our universe with a couple zeroes after,” he said.
“No, we’re talking magnitudes here, not simple multiplication factors,” was my answer.
And I clarified this by an upgrade of my old school teachers story.

“Imagine the matter of the entire universe brought together into a gigantic pile. Don’t bother about the gravity and it’s inclination to immediately compress the pile into the mother of all black holes, the gravity is temporarily switched off.”

“Now once in a million year our bird comes along to sharpen his beak on the pile of matter. And in doing so, he removes one elementary particle from our pile.
After another million years he is back and removes a second particle. And so on.”

“After the entire pile is gone by the bird’s actions, merely 1086 years went by.
When he repeats the whole job (in a parallel universe, because this one is now empty) a hundred trillion times (US counting system, thus 1014 times) then a 10100 years have passed.”
This indicates clearly that 10100 is a formidable number.
In decimal notation: 10. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000.
000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000.

In fact this number has it’s own name. It is called ‘googol’. This name was coined by the nephew of a mathematician, a 9-year old.
Of course someone came up with the even bigger number:10googol and called this googolplex. This can be written as the power tower , a number with googol digits.

This shows the efficiency of the power tower notation.
With only 3 digits written as a number with more than 360.000.000 digits is created.

There was a guy who manually calculated the exact number of digits of this small power tower. Speaking about fascination. But this is just playing with numbers of course. It’s not seriously applied mathematics.

This changed with the work of Samual Skewes. I made a small note about his efforts in an earlier post:
Skewes came up in 1933 with an upper-bound for a certain calculation as high as .

This is an insanely large number. G.H. Hardy said that it was the largest number ever used in a serious mathematical problem.
“Imagine a chessboard on which every elementary particle of the universe is placed on a private square,” he said. “Now swap places two by two. The game ends when all possible
swaps have been exhausted and the starting position is reached for the third time. Then the accumulated number of those moves is about equal to Skewes number.”

At that time it seemed to me that I could easily surpass such a number.
I would use some idle time to scribble a few thousand exponents on top of the power tower, ending up with the largest number ever seen by mankind.
Admittedly without any serious math underneath but still. My number would never be beaten.

Or so I thought.
Alas, I couldn’t be further off track. I stumbled upon Graham’s number.
To say that Skewes futile attempt is dwarfed by Graham’s number, is hopelessly anthropological that is trying to get it on a human scale.
There are no words in our language to describe the size of Graham.
What about expressing the diameter of the visible universe (9×1026 m) in Planck length units (1.6×10−35 m)?
Well that merely gives us 5,6×1061 units. Even Skewes can do better.

  • The universe is too small to write Graham’s number down even when the digits are of Planck length size.
  • The universe is too small to write the number of digits of Graham’s number down.
  • The universe is too small to write the number of digits of the number of digits of Graham’s number down.
  • The universe is too small to write the number of digits ……. of the number of digits of Graham’s number down.
  • And on. And on…

So what is going on?

Ronald Graham explored a problem in a peculiar part of mathematics, called Ramsey Theory.
The problem itself is utterly abstract but scaled back to the absolute minimum of words it reads as: what is the upper bound below which a certain phenomenon must occur in a multi dimensional system.
He derived an upper bound and coughed up his famous number that earned him a place in the Guinness Book of Records.

Because the number can’t be written as a power tower – the most powerful tool so-far – he had to use a special – by Donald Knuth – designed notation.
I don’t know if Knuth had this ready for use or did the design on Graham’s request.
Nevertheless, Ronald could start and he started in a familiar way.

In computer language we sometimes see 33 written as 3↑3, which is of course 27.

Now Knuth dictates that 3↑↑3 = 3↑(3↑3) = 327 = 7.625.597.484.987 a sizable number already.

I specifically point out that 3↑↑3 is the same as a power tower of 3’s.
In general: a↑↑b means a power tower of a’s, b elements high.
This is called tetration, which is iterated exponentiation.

Iteration is a well-known phenomenon in math of course.
Repeated addition became multiplication. 5+5+5+5+5+5 = 6×5
Repeated multiplication became exponentiation. 5x5x5x5 = 54.
Repeated exponentiation became tetration. a↑↑b is a power tower of a’s which is b high.

With this refreshed knowledge let us now zoom in on 3↑↑↑3. Also known as pentation, aka as iterated tetration.
We ‘factorize’ the expression as 3↑↑(3↑↑3). The expression in between the brackets is our familiar power tower and we know it’s value: 7.625.597.484.987
So we can write 3↑↑(3↑↑3) as 3↑↑ 7.625.597.484.987.

I must suppress the urge, to continue in capital letters: 3↑↑ 7.625.597.484.987 is a power tower of 3’s, 7.625.597.484.987 powers high!!! A power tower of more than seven trillion exponents!
What a number that must be! Are we approaching ‘Graham’ already?
No. Not yet. First we go up another iteration level. That level is called hexation.
We now have to deal with 3↑↑↑↑3.

I won’t even try to visualize the consequences of adding this 4th arrow.
Where pentation did explode the number of exponents in the power tower, hexation sprinkles power towers as if these were cheap candy. Hexation uses the final calculated value of the first power tower as the number of exponents in the next tower. And from that tower feeds the final value into the next. And on.

The thing to remember is: each next arrow opens a hyper-operation sequence of the previous operation.

Reading along about ‘The Number’ and just having digested the 3↑↑↑↑3 explosion and the dramatic effect of adding just another arrow, I expected Graham to put a couple of extra up-arrows for good measure and call it a day.
I mean, to the non-mathematician it must seem as if infinity comes into sight already, or at least is nearby.
Now his number had become so immense that he should end up straight in The Guinness Book right?
No, actually that’s not what happened.

What Graham did was taking 3↑↑↑↑3 as a starting point. He called this point G1.
Then he created G2 as 3↑↑…..↑↑3 where the number of arrows is 3↑↑↑↑3.

Maybe it is time for a brief pause. This is not merely mind numbing, the mind totally gives up here. Graham casually put in an unspeakable number of arrows while we’ve just seen what one added up-arrow can do.

Anyway, after that step, he went on with G3 where the number of arrows is equal to G2.
Ronald Graham continued this procedure and finally came to a stop at G64. He had created his Number. And had earned his entry in the Guinness Book of Records.


  1. The odd thing is that apparently, this outcome is the solution to a problem.
    But how can we imagine the problem, when the solution is totally beyond our comprehension?
    I have no clue.
    I guess it has something to do with the multi dimensional nature of the issue.
    Each higher dimension dramatically blows up its internal structure when extending the previous one.
  2. The thing with the human brain is that for very small and for very large numbers, it tries to make a reference to something more or less familiar. A proton, an amoeba, an ant, us, a whale, a tree, a town, a country, earth, the solar system, our galaxy, the universe.
    When a number totally outscales this reference system, we’re lost. We can try to extend this a bit with examples from probability theory like G.H. Hardy did with Skewes number.
    But in the case of the real giant numbers we’re lost. The mind cannot even start to
  3. And then I found this on the internet: “Now, there are larger numbers used in mathematics which have dethroned Graham, such as TREE(3), SCG(13), and other terms in far-reaching sequences.”
    It seems to me that these numbers are so big, that infinity lies halfway of them.

I asked ChatGPT for an explication of the meaning of Graham’s work, simple enough to be understood by the layman.
This is what he [ChatGPT] said:

Explication of ChatGPT.

The problem for which Graham’s number provides an upper bound is a question in the field of mathematics called Ramsey theory. Ramsey theory basically deals with finding a point of guaranteed order within a large enough system, even if it looks chaotic on the surface.
In this specific problem, you’re looking at a kind of network or a “graph”.

Picture a cube. It has 8 corners, and there are lines connecting the corners to each other.
Now, imagine you can color each of these lines either red or blue.

Graham’s number comes into play when we ask a question like this: What is the smallest number of corners the cube would have to have such that no matter how you color the lines, you can always find a flat square (or a “plane”) with four corners, all connected by lines of the same color?

It turns out that there definitely is such a number, but we don’t know exactly what it is.
What Graham and his collaborator Rothman did was provide an upper bound for this number – the highest it could possibly be. That upper bound is Graham’s number. It’s an unfathomably large number – so large that trying to picture it or write it out would be beyond comprehension.

So, in layman’s terms, Graham’s number is the largest possible answer to a problem about coloring lines connecting points and finding a smaller pattern within it. It doesn’t tell us the exact answer, but it lets us know the answer can’t be larger than Graham’s number.


  • The Penguin guide of curious and interesting numbers by David Wells
  • Wikipedia
  • And of course, ChatGPT


Eef, 2023-06

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