The largest number in the world is Graham. Graham's unimaginable number. Mersenne primes

epigraph
If you peer into the abyss for a long time,
you can have a good time.
Mechanical Soul Engineer

As soon as a child (and this happens around the age of three or four) understands that all numbers are divided into three groups “one, two and many,” he immediately tries to figure out: how much is too much, how a lot of differs from so many, and might it turn out so much that it doesn't happen anymore. Surely you played an interesting (for that age) game with your parents, who can name the largest number, and if the ancestor was no dumber than a fifth grader, then he always won, answering “two million” for every “million”, and “two billion” or “billion plus one” for every “billion”.

Already by the first grade of school, everyone knows that there are an infinite number of numbers, they never end, and there is no such thing as the largest number. To anyone million trillion billion You can always say “plus one” and still win. And a little later the understanding comes (should come!) that long strings of numbers by themselves mean nothing. All these trillions billions They make sense only when they serve as a representation of a certain number of objects or describe a certain phenomenon. It’s not difficult to come up with a long number that doesn’t represent anything other than a set of long-sounding numbers; infinite number. Science, to some figurative extent, is engaged in looking for very specific combinations of numbers in this vast abyss, adding to some physical phenomenon, such as the speed of light, Avogadro's number or Planck's constant.

And the question immediately arises, what is the largest number in the world that means something? In this article I will try to talk about the digital monster called Graham number, although strictly speaking, science knows larger numbers. Graham's number is the most hyped, one might say "heard" number among the general public, because it is quite simple to explain and yet large enough to turn heads. In general, here it is necessary to declare a small disclaimer ( rus. warning). It may sound like a joke, but I'm not joking at all. I say quite seriously - meticulous delving into such mathematical depths, coupled with the unrestrained expansion of the boundaries of perception, can (and will have) a serious impact on the worldview, on the positioning of the individual in society, and, ultimately, on general psychological state picking, or, let's call a spade a spade - opens the way to silliness. There is no need to read the following text too carefully, and you should not imagine the things described in it too vividly and vividly. And don’t say later that you weren’t warned!

Before moving on to monster numbers, let’s practice first on cats. Let me remind you that to describe large numbers (not monsters, but simply large numbers) it is convenient to use scientific or so-called. exponential recording method.

When they talk, say, about the number of stars in the Universe (in the Observable Universe), no idiot bothers to calculate how many there are literally, down to the last star. It is believed that there are approximately 10 21 pieces. And this is a lower estimate. This means that the total number of stars can be expressed by a number that has 21 zeros after the one, i.e. "1,000,000,000,000,000,000,000."

This is what a small fraction of them (about 100,000) in the Omega Centauri globular cluster looks like.

Naturally, when we are talking about such scales, the actual numbers in the number do not play a significant role, after all, everything is very conditional and approximately. May be In fact the number of stars in the Universe is “1,564,861,615,140,168,357,973,” or maybe “9,384,684,643,798,468,483,745.” Or even “3 333 333 333 333 333 333 333”, why not, although it’s unlikely, of course. In cosmology, the science of the properties of the Universe as a whole, one does not bother with such trifles. The main thing is to imagine that approximately this number consists of 22 digits, which makes it more convenient to consider it as one followed by 21 zeros, and write it as 10 21. The rule is general and very simple. Whatever figure or number stands in place of the degree (printed in small print on top of 10 here), how many zeros after the unit will be in this number, if you paint it in a simple way, with signs in a row, and not in a scientific way. Some numbers have "human names", for example we call 10 3 "thousand", 10 6 - "million", and 10 9 - "billion", but some do not. Let's say 10 59 does not have a generally accepted name. And 10 21, by the way, has it - this is a “sextillion”.

Everything that goes up to a million is intuitively understandable to almost any person, because who doesn't want to become a millionaire? Then some people start having problems. Although almost everyone knows a billion (10 9). You can even count to a billion. If, just after being born, literally at the moment of birth, you start counting once a second “one, two, three, four...” and don’t sleep, don’t drink, don’t eat, but just count, count, count tirelessly day and night, then when you turn 32, you can count to a billion, because 32 revolutions of the Earth around the Sun take about a billion seconds.

7 billion is the number of people on the planet. Based on the above, it is absolutely impossible to count them all in order during a human life; you will have to live more than two hundred years.

100 billion (10 11) - this is how many or so people have lived on the planet throughout its history. McDonald's sold 100 billion hamburgers by 1998 during its 50 years of existence. 100 billion stars (well, a little more) are in our galaxy Milky Way, and the Sun is one of them. The observable Universe contains the same number of galaxies. There are 100 billion neurons in the human brain. And the same number of anaerobic bacteria live in the cecum of everyone reading these lines.

Quadrillion (10 15, million billion) - that’s how many ants there are on the planet. This word normal people They don’t say it out loud, well, admit it, when was the last time you heard “a quadrillion of something” in a conversation?

Quintillion (10 18, billion billion) - this is how many possible configurations exist when solving a 3x3x3 Rubik's cube. Also the number of cubic meters of water in the world's oceans.

Sextillion (10 21) - we have already encountered this number. The number of stars in the Observable Universe. The number of grains of sand in all the deserts on Earth. The number of transistors in all existing electronic devices of mankind, if Intel did not lie to us.

10 sextillion (10 22) is the number of molecules in a gram of water.

10 24 is the mass of the Earth in kilograms.

10 26 is the diameter of the Observable Universe in meters, but counting in meters is not very convenient; the generally accepted boundaries of the Observable Universe are 93 billion light years.

To visualize cosmic proportions, it is useful to study this picture, expanding it to full screen.

However, even in the Observable Universe you can cram much more something other than meters.

10 51 atoms make up planet Earth.

10 80 approximate quantity elementary particles in the Observable Universe.

10 90 is the approximate number of photons in the Observable Universe. There are almost 10 billion times more of them than elementary particles, electrons and protons.

10 100 - googol. This number doesn't mean anything physically, it's just round and pretty. The company that set itself the goal of indexing Google's links (just kidding, of course, this is more than the number of elementary particles in the Universe!) in 1998 took the name Google.

10,122 protons will be needed to fill the Observable Universe to capacity, tightly, proton to proton, end to end.

The Observable Universe occupies 10,185 Planck volumes. Our science does not know smaller quantities than the Planck volume (a cube with a Planck length of 10–35 meters). Surely, as with the Universe, there is something even smaller there, but scientists have not yet come up with sane formulas for such trifles, it’s just pure speculation.

It turns out that 10,185 or so is the largest number that could, in principle, mean something in modern science. In a science that can touch and measure. It is something that exists or could exist if it happened that we had learned everything there was to know about the Universe. The number consists of 186 digits, here it is:

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Or have you heard about string theory, according to which there can be about 10,500 configurations of string vibrations, which means the same number of potential universes, each with its own laws.

I will only mention the well-known googolplex. A number that has googol digits, ten to the power of googol (10 googol), or ten to the power of ten to the power of one hundred (10 10 100).

10 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

I won't write it down in numbers. Googolplex means absolutely nothing. A person cannot imagine a googolplex of anything, it is physically impossible. To write down such a number, you will need the entire Observable Universe, if you write with a “nano-pen” directly across the vacuum, actually into the Planck cells of the cosmos. Let's convert all matter into ink and fill the Universe with just solid numbers, then we'll get a googolplex. But mathematicians ( scary people!) They are just warming up with googolprex, this is the lowest bar from which real nobodies start for them. And if you think that googolplex to the power of googolplex is what we're talking about, you have no idea HOW wrong you are.

After the googolplex there are many interesting numbers that have one role or another in mathematical proofs, but let’s go straight to the Graham number, named after (well, naturally) the mathematician Ronald Graham. First, I’ll tell you what it is and why it’s needed, then figuratively and on your fingers™ I’ll describe what its size is, and then I’ll write the number itself. More precisely, I will try to explain what I wrote.

Graham's number appeared in a paper devoted to solving one of the problems in Ramsey theory, and "Ramsey" is not a gerund here imperfect form, and the name of another mathematician, Frank Ramsey. The task, of course, is quite far-fetched from a layman’s point of view, although not very complicated, and even easily understandable.

Imagine a cube, all of whose vertices are connected by lines-segments of two colors, red or blue. Connected and colored in random order. Some people have already guessed that we will talk about a branch of mathematics called combinatorics.

Will we be able to contrive and choose a configuration of colors (and there are only two of them - red and blue) so that when coloring these segments we DO NOT end up with all the segments of the same color connecting the four vertices lying in the same plane? IN in this case, do NOT represent such a figure:

You can think about it yourself, spin the cube in your imagination before your eyes, it’s not so difficult to do this. There are two colors, the cube has 8 vertices (corners), which means there are 28 segments connecting them. You can choose the coloring configuration in such a way that we will not get the above figure anywhere, there will be multi-colored lines in all possible planes.

What if we have more dimensions? What if we take not a cube, but a four-dimensional cube, i.e. tesseract? Can we pull off the same trick we did with 3D?

I won’t even begin to explain what a four-dimensional cube is, does everyone know? U four-dimensional cube 16 peaks. And you don’t need to rack your brain and try to imagine a four-dimensional cube. This is pure mathematics. I looked at the number of dimensions, plugged it into the formula, and got the number of vertices, edges, faces, and so on. So a four-dimensional cube has 16 vertices and 120 segments connecting them. The number of coloring combinations in the four-dimensional case is much greater than in the three-dimensional case, but even here it is not very difficult to count, divide, reduce, and the like. In short, find out what's in four-dimensional space You can also get creative with coloring the segments of a hypercube in such a way that all lines of the same color connecting 4 vertices will not lie in the same plane.

In the fifth dimension? And in the fifth-dimensional, where the cube is called a penteract or pentacube, it is also possible.
And in the six-dimensional.

And then there are difficulties. Graham was unable to mathematically prove that a seven-dimensional hypercube could perform such an operation. Both eight-dimensional and nine-dimensional, and so on. But this “and so on”, it turned out, does not go to infinity, but ends with some very a large number, which is called the “Graham number”.

That is, there is some minimum dimension hypercube, in which the condition is violated, and it is no longer possible to avoid the combination of coloring of segments such that four points of the same color will lie in the same plane. And this minimum dimension is definitely more than six and definitely less number Graham, this is the scientist's mathematical proof.

Consider an n-dimensional hypercube and connect all pairs of vertices to obtain a complete graph with 2n vertices. Let's color each edge of this graph either red or blue. At what lowest value n does each such coloring necessarily contain a single-colored complete subgraph with four vertices, all of which lie in the same plane?

Many years have passed since the 70s, they were found math problems in which numbers appear larger than Graham’s, but this first monster number so amazed contemporaries, who understood the scale we are talking about, that in 1980 it was included in the Guinness Book of Records as “the most big number, ever involved in a rigorous mathematical proof" at that time.

Let's try to figure out how big it is. The largest number that can have any physical meaning 10 185, and if we fill the entire Observable Universe with a seemingly endless set of tiny numbers, we get something commensurate with googolplex.

Can you imagine this vastness? Forward, backward, up, down, as far as the eye can see and as far as the eye can see Hubble telescope, and even how much is missing, to the most distant galaxies and looking beyond them - numbers, numbers, numbers much smaller than a proton. Such a Universe, of course, will not be able to exist for long; it will immediately collapse into a black hole. Do you remember how much information can theoretically fit into the Universe? I am.

dochulion = 10 10 185

Let's open the doors of perception a little wider. Remember? That our Universe is just one of many bubbles in the Multiverse. And if you imagine dochulion such bubbles? Let's take a number as long as all that exists and imagine a Multiverse with a similar number of universes, each of which is covered to the brim with numbers - we get dochulion dokhulion. Can you imagine this? How you float in the non-existence of a scalar field, and all around you are universes-universes and in them numbers-numbers-numbers... I hope that such a nightmare (though, why a nightmare?) will not torment (and why torment?) an overly impressionable reader at night.

What am I getting at? Should you slow down? Come on, hoba, and one more flip! And now we have as many universes as there were numbers in the universes, the number of which was equal to up to a million numbers that filled our Universe. And immediately, without stopping, flip again. And the fourth and fifth. Tenth, thousandth. Do you keep up with your thoughts, can you still imagine the picture?

dohuliard = flip flips

We don’t stop and continue to flip dohullions of dohuliards as long as we have the strength. Until your eyes get dark, until you want to scream. Here everyone is their own brave Buratina, the safe word will be “cheese cheese”.

So here it is. What is this all about? The huge and infinite dohullions of flips and dohuliards of universes of complete digits cannot be compared to Graham's number. They don't even scrape the surface. If Graham's number is represented as a stick, stretched according to tradition throughout the entire Observable Universe, then we are here with you screwed up it will turn out to be a notch of thickness... well... how can I put it this way, to put it mildly... unworthy of mention. So, I softened it as best I could.

The number g 1 is equal to "three, four arrows, three." What does it mean? This is what a recording method called Knuth arrow notation looks like.

One arrow means ordinary exponentiation.

1010 = 10 10 = 10 000 000 000

Two arrows mean, clearly, raising to a power of a power.

23 = 222 = 2 2 2 = 2 4 = 16

33 = 333 = 3 3 3 = 3 27 = 7,625,597,484,987 (more than 7 trillion)

34 = 3333 = 3 3 3 3 = 3 7 625 597 484 987 = a number with about 3 trillion digits

35 = 33333 = 3 3 3 3 3 = 3 3 7 625 597 484 987 = 3 to the power of a number with 3 trillion digits - googolplex already sucks

5 5 5 5 5 5 5 5

Let's move on to the three arrows. If the double arrow showed the height of the tower of degrees, then the triple arrow would seem to indicate “the height of the tower of the height of the tower”? What the hell! In the case of three, we have the height of the tower the height of the tower the height of the tower (there is no such concept in mathematics, I decided to call it " crazy"). Something like this:

That is, 33 forms a crazy tower of triplets, 7 trillion high. What are 7 trillion threes stacked on top of each other and called "crazy"? If you carefully read this text and did not fall asleep at the very beginning, you probably remember that from Earth to Saturn there are 100 trillion centimeters. The three shown on the screen in the twelfth font, this one - 3 - is five millimeters high. This means that a crazy series of threes will stretch from your screen... well, not to Saturn, of course. It won’t even reach the Sun, only a quarter of an astronomical unit, about the distance from Earth to Mars in good weather. Please note (don’t sleep!) that recklessness is not a number the length from Earth to Mars, it’s tower of degrees so tall. We remember that five triplets in this tower cover the googolplex, calculating the first decimeter of triplets burns all the fuses of the planet’s computers, and the remaining millions of kilometers of degrees are, as it were, useless, they simply openly mock the reader, counting them is useless and impossible.

Now it’s clear that 34 = 3333 = 337 625 597 484 987 = 3 towerless, (not 3 to the degree of towerless, but “three arrows, towerless arrow”(!)), aka crazy crazy will not fit either in length or height into the Observable Universe, and will not even fit into the supposed Multiverse.

At 35 = 33333 the words end, and at 36 = 333333 the interjections end, but you can practice if you are interested.

It is useless to calculate it, and it won’t work. The number of degrees here cannot be meaningfully counted. This number is impossible to imagine, it is impossible to describe. No analogies on your fingers™ are not applicable, the number simply has nothing to compare with. We can say that it is huge, that it is grandiose, that it is monumental and that it looks beyond the horizon of events. That is, give it some verbal epithets. But visualization, even free and imaginative, is impossible. If with three arrows it was still possible to say something, to draw a recklessness from Earth to Mars, to somehow compare it with something, then there simply cannot be any analogies. Try to imagine a thin tower of triplets from Earth to Mars, next to another almost the same, and another, and another... An endless field of towers goes into the distance, into infinity, towers everywhere, towers everywhere. And what’s most offensive is that these towers don’t even have anything to do with the number, they only determine the height of other towers that need to be built in order to get the height of the towers, in order to get the height of the towers... so that after an unimaginable amount of time and iterations they get the number itself.

That's what g 1 is, that's what 33 is.

Have you rested? Now, from g 1, we return with renewed vigor to the assault on Graham’s number. Have you noticed how the escalation increases from arrow to arrow?

33 = 7 625 597 484 987

33 = tower, the height of the Earth to Mars.

33 = a number that is impossible to imagine or describe.

Can you imagine what kind of digital nightmare happens when the shooter turns out to be five? When are there six? Can you imagine the number when the shooter will be one hundred? If you can, let me bring to your attention the number g 2 in which the number of these arrows turns out to be equal to g 1. Remember what g 1 is, right?

I won’t hide it, there is also g 3, which contains g 2 arrows. By the way, is it still clear that g 3 is not g 2 “to the power” of g 2, but the number of crazy people that determine the height of crazy towers that determine the height... and so on along the entire chain down to the thermal death of the Universe? This is where you can start crying.

Why cry? Because that's absolutely true. There is also the number g 4, which contains g 3 arrows between the triplets. There is also g 5, there is g 6 and g 7 and g 17 and g 43...

In short, there are 64 of these g. Each previous one is numerically equal to the number of arrows in the next one. The last g 64 is Graham’s number, with which everything so seemingly innocently began. This is the number of dimensions of the hypercube, which will definitely be enough to correctly color the segments with red and blue colors. Maybe less, this is, so to speak, the upper limit. It is written as follows:

And they write it like this:

That's it, now you can honestly relax. There is no longer any need to imagine or calculate anything. If you have read this far, everything should already fall into place. Or don't get up. Or not on your own.

Yes, an experienced reader with pumped fuses, no need for reproaches, you are absolutely right. Graham's number is a far-fetched and made-up bullshit. All these dimensionless hypercubes and abstract planes, damn them, who needs them? Where are the kilograms, where are the electrons, where is what can be measured? What kind of empty ranting about nothing? I agree. We can say that today's post on your fingers™ as far as possible, it is far from real science, almost all of it floats in some abstruse mathematical fantasies, while scientists do not have enough money for instruments, the world energy problem has not been solved, and someone still has a toilet in the courtyard. And who has it in the field?

But you know, there is such a theory, also very ephemeral and philosophical, you may have heard - everything that a person could imagine or imagine will definitely come true someday. Because the development of a civilization is determined by the extent to which it was able to translate the fantasies of the past into reality.

Nobody knows what the future holds for us. Human civilization has a thousand ways to end: nuclear wars, ecological disasters, deadly pandemics, whatever asteroid might arrive, dinosaurs won’t let you lie. The development of humanity can stop by itself, suddenly there is such a law that upon reaching a certain level, development simply stops and that’s it. Or representatives of the intergalactic union will arrive and stop this development by force.

But there is still, and not a small, chance that the development of humanity will continue without stopping. Even if it is not as dizzyingly fast as in the last 100 years, the main thing is that it is moving forward, the main thing is that it is progressive.

Nature has one unshakable law, known to us since ancient times. No matter what happens, no matter what we think to ourselves, time will not go away, it will pass. Whether we want it or not, with or without us, a thousand and 10 thousand years will pass.

200 years ago, a flying carpet (ordinary airplane), a magic mirror (Skype video) or the distant kingdom (the surface of the planet Mars) seemed like a pipe dream; 2,000 years ago, they relied only on the gods; 20,000 years ago, they couldn’t imagine this at all; there was not enough imagination . Can you say what will be available to people in 200 years? In 2000, in 20,000 years?

Will humanity survive, will it even be humanity with the prefix “human-”, or maybe by that time the stage of Artificial Intelligence will end, giving rise to some kind of ethereal energy substances of a special category of awareness? Maybe yes maybe no.

What if a million years pass? But he will go wherever he goes. Graham's number, and in general everything that a person is able to think about, imagine, pull out of oblivion and make, if not tangible, but at least an entity that has some meaning, will definitely come to fruition sooner or later. Just because we had enough strength today develop to the ability to realize such.

Today, tomorrow, when you have the opportunity, throw your head back into the night sky. Remember that moment of feeling your own insignificance? Do you feel how tiny a person is? A speck of dust, an atom compared to the boundless Universe, which is full of countless stars, and the abyss, accordingly, is not small either.

Next time, try to feel how the Universe is a grain of sand compared to what is happening in your head. What abyss opens up, what immeasurable concepts are born, what worlds are built, how the Universe flips inside out with just one movement of thought, how and how living, intelligent matter differs from dead and irrational matter.

I believe that after some time a person will reach out to Graham's number, touch it with his hand, or whatever he will have instead of a hand by then. This is not a valid, scientifically proven thought, it is really just a hope, something that inspires me. Not Faith with a capital F, not religious ecstasy, not doctrine and not spiritual practice. This is what I expect from humanity. I strive to help, to the best of my ability. Although, out of caution, I continue to classify myself as an agnostic.

There are numbers that are so incredibly, incredibly large that it would take the entire universe to even write them down. But here's what's really crazy... some of these unfathomably large numbers are crucial to understanding the world.

When I say “the largest number in the universe,” I really mean the largest significant number, the maximum possible number that is useful in some way. There are many contenders for this title, but I'll warn you right away: there really is a risk that trying to understand it all will blow your mind. And besides, with too much math, you won't have much fun.

Googol and googolplex

Edward Kasner

We could start with what are quite possibly the two biggest numbers you've ever heard of, and these are indeed the two biggest numbers that have generally accepted definitions in English language. (There is a fairly precise nomenclature used to denote numbers as large as you would like, but these two numbers you will not find in dictionaries nowadays.) Googol, since it became world famous (albeit with errors, note. in fact it is googol) in the form of Google, born in 1920 as a way to get children interested in big numbers.

To this end, Edward Kasner (pictured) took his two nephews, Milton and Edwin Sirott, for a walk through the New Jersey Palisades. He invited them to come up with any ideas, and then nine-year-old Milton suggested “googol.” Where he got this word from is unknown, but Kasner decided that or a number in which one hundred zeros follow the unit will henceforth be called a googol.

But young Milton did not stop there; he proposed an even larger number, the googolplex. This is a number, according to Milton, in which the first place is 1, and then as many zeros as you could write before you got tired. While the idea is fascinating, Kasner decided a more formal definition was needed. As he explained in his 1940 book Mathematics and the Imagination, Milton's definition leaves open the risky possibility that an accidental buffoon could become a mathematician superior to Albert Einstein simply because he has greater stamina.

So Kasner decided that a googolplex would be , or 1, and then a googol of zeros. Otherwise, and in notation similar to that which we will deal with for other numbers, we will say that a googolplex is . To show how fascinating this is, Carl Sagan once noted that it is physically impossible to write down all the zeros of a googolplex because there simply isn't enough space in the universe. If we fill the entire volume of the observable Universe with small dust particles approximately 1.5 microns in size, then the number of different ways these particles can be arranged will be approximately equal to one googolplex.

Linguistically speaking, googol and googolplex are probably the two largest significant numbers (at least in the English language), but, as we will now establish, there are infinitely many ways to define “significance.”

Real world

If we talk about the largest significant number, there is a reasonable argument that this really means that we need to find the largest number with a value that actually exists in the world. We can start with the current human population, which is currently around 6920 million. World GDP in 2010 was estimated to be around $61,960 billion, but both of these numbers are insignificant compared to the approximately 100 trillion cells that make up the human body. Of course, none of these numbers can compare to the total number of particles in the Universe, which is generally considered to be approximately , and this number is so large that our language has no word for it.

We can play a little with the systems of measures, making the numbers larger and larger. Thus, the mass of the Sun in tons will be less than in pounds. A great way to do this is to use the Planck system of units, which are the smallest possible measures for which the laws of physics still apply. For example, the age of the Universe in Planck time is about . If we return to the first unit of Planck time after Big Bang, then we will see that the density of the Universe was then . We're getting more and more, but we haven't even reached googol yet.

The largest number with any real world application - or in this case real application in worlds - probably , - one of the latest estimates of the number of universes in the multiverse. This number is so large that the human brain will literally not be able to perceive all these different universes, since the brain is only capable of approximately configurations. In fact, this number is probably the largest number that makes any practical sense unless you take into account the idea of the multiverse as a whole. However, there are still much larger numbers lurking there. But to find them we must go into the realm of pure mathematics, and there is no better place to start than prime numbers.

Mersenne primes

Part of the challenge is coming up with a good definition of what a “significant” number is. One way is to think in terms of prime and composite numbers. A prime number, as you probably remember from school mathematics, is any natural number(note not equal to one), which is divisible only by and itself. So, and are prime numbers, and and are composite numbers. This means that any composite number can ultimately be represented by its own simple divisors. In some ways, the number is more important than, say, , because there is no way to express it in terms of the product of smaller numbers.

Obviously we can go a little further. , for example, is actually just , which means that in a hypothetical world where our knowledge of numbers is limited to , a mathematician can still express the number . But the next number is prime, which means that the only way to express it is to directly know about its existence. This means that the largest known prime numbers play important role, but, say, a googol - which is ultimately just a set of numbers and , multiplied together - actually not. And since prime numbers are basically random, there is no known way to predict that an incredibly large number will actually be prime. To this day, discovering new prime numbers is a difficult undertaking.

Mathematicians Ancient Greece had a concept of prime numbers at least as early as 500 BC, and 2000 years later people still knew which numbers were prime only up to about 750. Thinkers in Euclid's time saw the possibility of simplification, but until the Renaissance mathematicians could not really put it into practice. These numbers are known as Mersenne numbers, named after the 17th century French scientist Marin Mersenne. The idea is quite simple: a Mersenne number is any number of the form . So, for example, , and this number is prime, the same is true for .

It is much faster and easier to determine Mersenne primes than any other kind of prime number, and computers have been hard at work searching for them for the past six decades. Until 1952, the largest known prime number was a number—a number with digits. In the same year, the computer calculated that the number is prime, and this number consists of digits, which makes it much larger than a googol.

Computers have been on the hunt ever since, and currently the Mersenne number is the largest prime number known to mankind. Discovered in 2008, it amounts to a number with almost millions of digits. It is the largest known number that cannot be expressed in terms of any smaller numbers, and if you want help finding an even larger Mersenne number, you (and your computer) can always join the search at http://www.mersenne. org/.

Skewes number

Stanley Skews

Let's look at prime numbers again. As I said, they behave fundamentally wrong, meaning that there is no way to predict what the next prime number will be. Mathematicians have been forced to resort to some pretty fantastic measurements to come up with some way to predict future prime numbers, even in some nebulous way. The most successful of these attempts is probably the prime number counting function, which was invented in the late 18th century by the legendary mathematician Carl Friedrich Gauss.

I'll spare you the more complicated math - we have a lot more to come anyway - but the gist of the function is this: for any integer, you can estimate how many prime numbers there are that are smaller than . For example, if , the function predicts that there should be prime numbers, if there should be prime numbers smaller than , and if , then there should be smaller numbers that are prime.

The arrangement of the prime numbers is indeed irregular and is only an approximation of the actual number of prime numbers. In fact, we know that there are prime numbers less than , prime numbers less than , and prime numbers less than . This is an excellent estimate, to be sure, but it is always only an estimate... and, more specifically, an estimate from above.

In all known cases up to , the function that finds the number of primes slightly overestimates the actual number of primes smaller than . Mathematicians once thought that this would always be the case, ad infinitum, and that this would certainly apply to some unimaginably huge numbers, but in 1914 John Edensor Littlewood proved that for some unknown, unimaginably huge number, this function would begin to produce fewer primes, and then it will switch between the top estimate and the bottom estimate an infinite number of times.

The hunt was for the starting point of the races, and then Stanley Skewes appeared (see photo). In 1933, he proved that the upper limit when a function approximating the number of prime numbers first produces a smaller value is the number . It is difficult to truly understand even in the most abstract sense what this number actually represents, and from this point of view it was the largest number ever used in a serious mathematical proof. Mathematicians have since been able to reduce the upper bound to a relatively small number, but the original number remains known as the Skewes number.

So how big is the number that dwarfs even the mighty googolplex? In The Penguin Dictionary of Curious and Interesting Numbers, David Wells recounts one way in which the mathematician Hardy was able to conceptualize the size of the Skuse number:

“Hardy thought it was “the largest number ever served for any particular purpose in mathematics,” and suggested that if a game of chess were played with all the particles of the Universe as pieces, one move would consist of swapping two particles, and the game would stop when the same position was repeated a third time, then the number of all possible games would be approximately equal to Skuse's number.'

One last thing before we move on: we talked about the smaller of the two Skewes numbers. There is another Skuse number, which the mathematician discovered in 1955. The first number is derived from the fact that the so-called Riemann hypothesis is true - this is a particularly difficult hypothesis in mathematics that remains unproven, very useful when it comes to prime numbers. However, if the Riemann hypothesis is false, Skuse found that the starting point of the jumps increases to .

Problem of magnitude

Before we get to the number that makes even the Skewes number look tiny, we need to talk a little about scale, because otherwise we have no way of assessing where we're going to go. First let's take a number - it's a tiny number, so small that people can actually have an intuitive understanding of what it means. There are very few numbers that fit this description, since numbers greater than six cease to be separate numbers and become “several”, “many”, etc.

Now let's take , i.e. . Although we actually cannot intuitively, as we did for the number, understand what it is, it is very easy to imagine what it is. So far so good. But what happens if we move to ? This is equal to , or . We are very far from being able to imagine this quantity, like any other very large one - we lose the ability to comprehend individual parts somewhere around a million. (Admittedly, it would take an insanely long time to actually count to a million of anything, but the point is that we are still capable of perceiving that number.)

However, although we cannot imagine, we are at least able to understand general outline, what is 7600 billion, perhaps comparing it to something like US GDP. We have moved from intuition to representation to simple understanding, but at least we still have some gap in our understanding of what a number is. That's about to change as we move another rung up the ladder.

To do this, we need to move to a notation introduced by Donald Knuth, known as arrow notation. This notation can be written as . When we then go to , the number we get will be . This is equal to where the total of threes is. We have now far and truly surpassed all the other numbers we have already talked about. After all, even the largest of them had only three or four terms in the indicator series. For example, even the super-Skuse number is “only” - even with the allowance for the fact that both the base and the exponents are much larger than , it is still absolutely nothing compared to the size of a number tower with a billion members.

Obviously, there is no way to comprehend such huge numbers... and yet, the process by which they are created can still be understood. We couldn't understand the real quantity that is given by a tower of powers with a billion triplets, but we can basically imagine such a tower with many terms, and a really decent supercomputer would be able to store such towers in memory even if it couldn't calculate their actual values .

This is becoming more and more abstract, but it will only get worse. You might think that a tower of degrees whose exponent length is equal (indeed, in the previous version of this post I made exactly this mistake), but it is simple. In other words, imagine being able to calculate the exact value of a power tower of triplets that is made up of elements, and then you took that value and created a new tower with as many in it as... that gives .

Repeat this process with each subsequent number ( note starting from the right) until you do it times, and then finally you get . This is a number that is simply incredibly large, but at least the steps to get it seem understandable if you do everything very slowly. We can no longer understand the numbers or imagine the procedure by which they are obtained, but at least we can understand the basic algorithm, only in a long enough time.

Now let's prepare the mind to really blow it.

Graham number (Graham)

Ronald Graham

This is how you get Graham's number, which holds a place in the Guinness Book of World Records as the largest number ever used in a mathematical proof. It is absolutely impossible to imagine how big it is, and equally difficult to explain exactly what it is. Basically, Graham's number appears when dealing with hypercubes, which are theoretical geometric shapes with more than three dimensions. Mathematician Ronald Graham (see photo) wanted to find out at what smallest number of dimensions certain properties of a hypercube would remain stable. (Sorry for such a vague explanation, but I'm sure we all need to get at least two academic degrees in mathematics to make it more accurate.)

In any case, the Graham number is an upper estimate of this minimum number of dimensions. So how big is this upper bound? Let's return to the number, so large that we can only vaguely understand the algorithm for obtaining it. Now, instead of just jumping up one more level to , we will count the number that has arrows between the first and last three. We are now far beyond even the slightest understanding of what this number is or even what we need to do to calculate it.

Now let's repeat this process once ( note at each next step we write the number of arrows, equal to the number obtained in the previous step).

This, ladies and gentlemen, is Graham's number, which is about an order of magnitude higher than the point human understanding. It is a number that is so much greater than any number you can imagine—it is so much greater than any infinity you could ever hope to imagine—it simply defies even the most abstract description.

But here's a strange thing. Since the Graham number is basically just triplets multiplied together, we know some of its properties without actually calculating it. We can't represent the Graham number using any familiar notation, even if we used the entire universe to write it down, but I can tell you the last twelve digits of the Graham number right now: . And that's not all: we know at least the last digits of Graham's number.

Of course, it's worth remembering that this number is only an upper bound in Graham's original problem. It is quite possible that the actual number of measurements required to achieve the desired property is much, much less. In fact, it has been believed since the 1980s, according to most experts in the field, that there are actually only six dimensions—a number so small that we can understand it intuitively. The lower bound has since been raised to , but there is still a very good chance that the solution to Graham's problem does not lie anywhere near a number as large as Graham's number.

Towards infinity

So are there numbers greater than Graham's number? There is, of course, for starters there is the Graham number. As for the significant number... well, there are some fiendishly complex areas of mathematics (particularly the area known as combinatorics) and computer science in which numbers even larger than Graham's number occur. But we have almost reached the limit of what I can hope will ever be rationally explained. For those foolhardy enough to go even further, further reading is suggested at your own risk.

Well, now an amazing quote that is attributed to Douglas Ray ( note Honestly, it sounds pretty funny:

“I see clusters of vague numbers that are hidden there in the darkness, behind the small spot of light that the candle of reason gives. They whisper to each other; conspiring about who knows what. Perhaps they don't like us very much for capturing their little brothers in our minds. Or perhaps they simply lead a single-digit life, out there, beyond our understanding.

The largest mathematical constant
It's hard to imagine Infinity correctly without first imagining really big numbers. I'm not talking about tiny numbers that differ little from zero, such as the number of atoms in the universe or the number of years it would take a monkey to completely copy Shakespeare's works. I invite you to consider what was, around 1977, the largest number ever used in a serious mathematical proof. This proof, performed by Ronald Graham, provides an upper bound on the answers to a certain question in Ramsey's theory. In order to understand the proof, we need to introduce a new concept from Donald Knuth's work "the study of finite numbers." This concept is usually represented by a small upward-pointing arrow, which we'll label here as ^

3^3 = 3 * 3 * 3 = 27. This number is small enough to imagine.

3^^3 = 3^(3^3) = 3^27 = 7,625,597,484,987. More than 27, but small enough that I could print it. Nobody can imagine seven trillion, but we can easily understand this number, which roughly corresponds in order to the volume of GDP.

3^^^3 = 3^^(3^^3) = 3^(3^(3^(3^...^(3^3)...))). The interval "..." consists of 7,625,597,484,987 triplets. In other words, 3^^^3 or the arrow (3, 3, 3) is an exponential tower of triplets 7,625,597,484,987 levels high. This number is beyond human comprehension, but the procedure for creating it can be visualized. Let's take x=1. Set x to 3^x. Repeat this seven trillion times. Although the earliest stages of this number are too large to be contained in an entire universe, the exponential tower itself, written as "3^3^3^3...^3" is small enough to be contained in a modern supercomputer.

3^^^^3 = 3^^^(3^^^3) = 3^^(3^^(3^^...^^(3^^3)...)). Now both the number and the procedure for its creation are beyond human ability to conceive, although the procedure can be understood. Take x=1. Assign x the value of an exponential tower of length x. Repeat this 3^^^3 times, which is equal to an exponential tower of seven trillion triplets.

And the result is, in the words of Martin Gardner, “3^^^^3 is inconceivably larger than 3^^^3, but it is still small since most finite numbers are larger.”

And then Graham's number. Let x be equal to 3^^^^3, the inconceivably large number described above. Then assign x the value 3^^^^^^^(x arrow)^^^^^^^3. Do the same thing again, but replace x with (3^^^^^^^(x arrow)^^^^^^^3) Repeat this 63 times or 64 times, taking into account the initial sequence 3^^^^3.

Graham's number is far beyond my ability to comprehend. I can describe it, but I cannot perceive it correctly. (Perhaps Graham can accept it since he wrote a mathematical proof using it). This number is much larger than most people's concept of infinity. I know it was bigger than my imagination.

The real answer to Ramsey's problem, which gave rise to this number as an upper bound, was probably the number 6.

P.s In addition to my superstitious horror, this number gave rise to a little joke: Onotole Wasserman easily squares Graham’s number in a couple of seconds.

What is the biggest number in the world that means something? In this article I will try to talk about a digital monster called the Graham number.

Writes sly2m.livejournal.com

Source:

If you stare into the abyss for a long time, you can have a good time.
Mechanical Soul Engineer

Graham Finger Number™

As soon as a child (and this happens around the age of three or four) understands that all numbers are divided into three groups “one, two and many”, he immediately tries to find out: how much is a lot, how a lot differs from a lot, and whether there may be so many that there is no more. Surely you played an interesting (for that age) game with your parents, who can name the largest number, and if your ancestor was no dumber than a fifth-grader, then he always won, answering “two million” for every “million”, and “two million” for “billion” - "two billion" or "billion plus one".

Already by the first grade of school, everyone knows that there are an infinite number of numbers, they never end, and there is no such thing as the largest number. For any million trillion billion, you can always say “plus one” and still win. And a little later the understanding comes (should come!) that long strings of numbers by themselves mean nothing. All these trillions of billions only make sense when they serve as a representation of a certain number of objects or describe a certain phenomenon. There is no difficulty in coming up with a long number that represents nothing other than a set of long-sounding numbers; there are already an infinite number of them. Science, to some figurative extent, is engaged in looking for very specific combinations of numbers in this vast abyss, adding them to some physical phenomenon, for example, the speed of light, Avogadro’s number or Planck’s constant.

And the question immediately arises, what is the largest number in the world that means something? In this article I will try to talk about the digital monster called the Graham number, although strictly speaking, science knows larger numbers. Graham's number is the most hyped, one might say "heard" number among the general public, because it is quite simple to explain and yet large enough to make heads turn. In general, here it is necessary to declare a small disclaimer (Russian warning). It may sound like a joke, but I'm not joking at all. I say quite seriously - meticulous delving into such mathematical depths, combined with the unbridled expansion of the boundaries of perception, can have (and will have) a serious impact on the worldview, on the positioning of the individual in society, and, ultimately, on the general psychological state of the tinkerer, or, let’s call it things by their proper names - opens the way to silliness. There is no need to read the following text too carefully, and you should not imagine the things described in it too vividly and vividly. And don’t say later that you weren’t warned!

Before moving on to monster numbers, let’s first practice on cats. Let me remind you that to describe large numbers (not monsters, but simply large numbers) it is convenient to use scientific or so-called. exponential notation.

When they talk, say, about the number of stars in the Universe (in the Observable Universe), no idiot bothers to calculate how many there are literally, down to the last star. It is believed that there are approximately 10²¹ pieces. And this is a lower estimate. This means that the total number of stars can be expressed by a number that has 21 zeros after the one, i.e. "1,000,000,000,000,000,000,000."

This is what a small fraction of them (about 100,000) in the Omega Centauri globular cluster looks like.

Naturally, when we are talking about such scales, the actual numbers in the number do not play a significant role, after all, everything is very conditional and approximately. The actual number of stars in the Universe may be “1,564,861,615,140,168,357,973,” or maybe “9,384,684,643,798,468,483,745.” Or even “3 333 333 333 333 333 333 333”, why not, although unlikely, of course. In cosmology, the science of the properties of the Universe as a whole, one does not bother with such trifles. The main thing is to imagine that approximately this number consists of 22 digits, which makes it more convenient to consider it as one followed by 21 zeros, and write it as 10²¹. The rule is general and very simple. Whatever figure or number stands in place of the degree (printed in small print above 10), so many zeros after the unit will be in this number, if you paint it in a simple way, with signs in a row, and not in a scientific way. Some numbers have “human names”, for example we call 10³ “thousand”, 10⁶ “million”, and 10⁹ “billion”, but some do not. Let's say 10⁵⁹ does not have a generally accepted name. And 10²¹, by the way, has it - it’s a “sextillion”.

Everything that goes up to a million is intuitively clear to almost any person, because who doesn’t want to become a millionaire? Then some people start having problems. Although almost everyone knows a billion (10⁹). You can even count to a billion. If, just after being born, literally at the moment of birth, you start counting once a second “one, two, three, four...” and don’t sleep, don’t drink, don’t eat, but just count, count, count tirelessly day and night, then when you turn 32, you can count to a billion, because 32 revolutions of the Earth around the Sun take about a billion seconds.

7 billion is the number of people on the planet. Based on the above, it is absolutely impossible to count them all in order during a human life; you will have to live more than two hundred years.

100 billion (10¹¹) - this is how many or so people have lived on the planet throughout its history. McDonald's sold 100 billion hamburgers by 1998 during its 50-year existence. There are 100 billion stars (well, a little more) in our Milky Way galaxy, and the Sun is one of them. The observable Universe contains the same number of galaxies. There are 100 billion neurons in the human brain. And the same number of anaerobic bacteria live in the cecum of everyone reading these lines.

Trillion (10¹²) is a number that is rarely used. It is impossible to count to a trillion; it will take 32 thousand years. A trillion seconds ago, people lived in caves and hunted mammoths with spears. Yes, a trillion seconds ago mammoths lived on Earth. There are approximately a trillion fish in the planet's oceans. Our neighboring Andromeda galaxy has about a trillion stars. A person is made up of 10 trillion cells. Russia's GDP in 2013 amounted to 66 trillion rubles (in 2013 rubles). From Earth to Saturn, 100 trillion centimeters and the same number of letters in total were printed in all books ever published.

A quadrillion (10¹⁵, million billion) is the number of ants on the planet. Normal people don’t say this word out loud, well, admit it, when was the last time you heard “a quadrillion of something” in a conversation?

Quintillion (10¹⁸, billion billion) - this is how many possible configurations exist when solving a 3x3x3 Rubik's cube. Also the number of cubic meters of water in the world's oceans.

Sextillion (10²¹) - we have already encountered this number. The number of stars in the Observable Universe. The number of grains of sand in all the deserts on Earth. The number of transistors in all existing electronic devices of mankind, if Intel did not lie to us.

10 sextillion (10²²) is the number of molecules in a gram of water.

10²⁴ - mass of the Earth in kilograms.

10²⁶ is the diameter of the Observable Universe in meters, but counting in meters is not very convenient; the generally accepted boundaries of the Observable Universe are 93 billion light years.

To visualize cosmic proportions, it is useful to study this picture, expanding it to full screen.

However, even in the Observable Universe you can cram much more something other than meters.

10⁵¹ atoms make up planet Earth.

10⁸⁰ is the approximate number of elementary particles in the Observable Universe.

10⁹⁰ is the approximate number of photons in the Observable Universe. There are almost 10 billion times more of them than elementary particles, electrons and protons.

10¹⁰⁰ - googol. This number doesn't mean anything physically, it's just round and pretty. The company that set itself the goal of indexing Google's links (just kidding, of course, this is more than the number of elementary particles in the Universe!) in 1998 took the name Google.

10¹²² protons will be needed to fill the Observable Universe to capacity, tightly, proton to proton, end to end.

10¹⁸⁵ Planck volumes are occupied by the Observable Universe. Our science does not know smaller quantities than the Planck volume (a cube with a Planck length of 10⁻³⁵ meters). Surely, as with the Universe, there is something even smaller there, but scientists have not yet come up with sane formulas for such trifles, it’s just pure speculation.

It turns out that 10¹⁸⁵ or so is the largest number that, in principle, can mean something in modern science. In a science that can touch and measure. It is something that exists or could exist if it happened that we had learned everything there was to know about the Universe. The number consists of 186 digits, here it is:

Or have you heard about string theory, according to which there can be about 10⁵⁰⁰ configurations of string vibrations, which means the same number of potential universes, each with its own laws.

I’ll just mention the googolplex, which is well-known to many. A number with googol digits, ten to the power of googol or ten to the power of ten to the power of one hundred

I won't write it down in numbers. Googolplex means absolutely nothing. A person cannot imagine a googolplex of anything, it is physically impossible. To write down such a number, you will need the entire Observable Universe, if you write with a “nano-pen” directly across the vacuum, actually into the Planck cells of the cosmos. Let's convert all matter into ink and fill the Universe with just solid numbers, then we'll get a googolplex. But mathematicians (terrible people!) are just warming up with Googolprex, this is the lowest bar from which real good things start for them. And if you think that googolplex to the power of googolplex is what we're talking about, you have no idea HOW wrong you are.

After the googolplex there are many interesting numbers that have one role or another in mathematical proofs, but let’s go straight to the Graham number, named after (well, naturally) the mathematician Ronald Graham. First, I’ll tell you what it is and what it’s needed for, after which I’ll figuratively and on my fingers™ describe its size, and then I’ll write the number itself. More precisely, I will try to explain what I wrote.

Graham's number appeared in a paper devoted to solving one of the problems in Ramsey theory, and “Ramsey” here is not an imperfect gerund, but the surname of another mathematician, Frank Ramsey. The task, of course, is quite far-fetched from a layman’s point of view, although not very complicated, and even easily understandable.

What if we have more dimensions? What if we take not a cube, but a four-dimensional cube, i.e. tesseract? Can we pull off the same trick we did with 3D?

In the fifth dimension? And in the fifth-dimensional, where the cube is called a penteract or pentacube, it is also possible.
And in the six-dimensional.

And then there are difficulties. Graham was unable to mathematically prove that a seven-dimensional hypercube could perform such an operation. Both eight-dimensional and nine-dimensional, and so on. But this “and so on,” it turned out, does not go to infinity, but ends with some very large number, which was called the “Graham number.”

That is, there is some minimum dimension of the hypercube at which the condition is violated, and it is no longer possible to avoid the combination of coloring of segments such that four points of the same color will lie in the same plane. And this minimum dimension is definitely more than six and definitely less than Graham’s number, this is the scientist’s mathematical proof.

Consider an n-dimensional hypercube and connect all pairs of vertices to obtain a complete graph with 2n vertices. Let's color each edge of this graph either red or blue. For what is the smallest value of n, each such coloring necessarily contains a single-colored complete subgraph with four vertices, all of which lie in the same plane?

Many years have passed since the 70s, mathematical problems have been found in which numbers larger than Graham appear, but this first monster number so amazed contemporaries who understood the scale we are talking about that in 1980 it was included in the Guinness Book of Records as “the most the largest number ever involved in a rigorous mathematical proof” at that time.

Let's try to figure out how big it is. The largest number that can have any physical meaning is 10¹⁸⁵, and if the entire Observable Universe is filled with a seemingly endless set of tiny numbers, we get something comparable to a googolplex.

Can you imagine this vastness? Forward, backward, up, down, as far as the eye can see and as far as the Hubble telescope can see, and even as far as the Hubble telescope can, to the most distant galaxies and looking beyond them - numbers, numbers, numbers much smaller than a proton. Such a Universe, of course, will not be able to exist for long; it will immediately collapse into a black hole. Do you remember how much information can theoretically fit into the Universe?

The number is really huge, it blows your mind. It is not exactly equal to the googolplex, and it does not have a name, so I will call it “dochulion”. Just thought of it, why not. The number of Planck cells in the Observable Universe, and each cell contains a number. The number contains 10¹⁸⁵ digits and can be represented as

Let's open the doors of perception a little wider. Remember inflation theory? That our Universe is just one of many bubbles in the Multiverse. What if you imagine a dozen such bubbles? Let's take a number as long as everything that exists and imagine a Multiverse with a similar number of universes, each of which is covered to capacity with numbers - we get a dochulion of dochulions. Can you imagine this? How you float in the non-existence of a scalar field, and all around you are universes-universes and in them numbers-numbers-numbers... I hope that such a nightmare (though, why a nightmare?) will not torment (and why torment?) an overly impressionable reader at night.

For convenience, we will call this operation “flip”. Such a frivolous interjection, as if they took the Universe and turned it inside out, then it was inside in numbers, but now, on the contrary, we have as many universes outside as there were numbers, and each box is full, itself all in numbers. Just like you peel a pomegranate, you bend the crust, the grains turn out from the inside, and in the grains there are again pomegranates. I also came up with the idea on the fly, why not, it was a great ride with the dokhulion.

What am I getting at? Should you slow down? Come on, hoba, and one more flip! And now we have as many universes as there were numbers in the universes, the number of which was equal to up to a million numbers that filled our Universe. And immediately, without stopping, flip again. And the fourth and fifth. Tenth, thousandth. Do you keep up with your thoughts, can you still imagine the picture?

Let's not waste time on trifles, let's spread the wings of imagination, accelerate to the fullest and flip flip flips. We turn each universe inside out as many times as how many dozens of universes there were in the previous flip, which was a flip from the one before last, which... uh... well, are you following? Somewhere like this. Let our number now become, let’s say, “dohuliard”.

Dohuliard = flip of flips

We don’t stop and continue to flip dohullions of dohuliards as long as we have the strength. Until your eyes get dark, until you want to scream. Here everyone is their own brave Pinocchio, the safe word will be “cheese cheese”.

So here it is. What is this all about? The huge and infinite dohullions of flips and dohuliards of universes of complete digits cannot be compared to Graham's number. They don't even scrape the surface. If Graham's number is represented as a stick, traditionally stretched across the entire Observable Universe, then what we've come up with here will turn out to be a notch of thickness... well... how can I put it mildly... unworthy of mention. So, I softened it as best I could.

Now let's take a break and take a break. We read, we counted, our little eyes were tired. Let’s forget about Graham’s number, it’s still a long way to go, let’s unfocus our eyes, relax, meditate on a much smaller, even miniature number, which we’ll call g₁, and write it down in just six characters:
g₁ = 33

The number g₁ is equal to “three, four arrows, three.” What does it mean? This is what the writing method called Knuth's arrow notation looks like.

One arrow means ordinary exponentiation.

44 = 4⁴ = 256

1010 = 10¹⁰ = 10,000,000,000

Two arrows mean, clearly, raising to a power of a power.

In short, “number arrow arrow another number” shows what height of powers (mathematicians say “tower”) is built from the first number. For example, 58 means a tower of eight fives and is so large that it cannot be calculated on any supercomputer, even on all computers on the planet at the same time.

That is, 33 forms a crazy tower of triplets, 7 trillion high. What are 7 trillion threes stacked on top of each other and called “crazy”? If you carefully read this text and did not fall asleep at the very beginning, you probably remember that from Earth to Saturn there are 100 trillion centimeters. The three shown on the screen in the twelfth font, this one - 3 - is five millimeters high. This means that a crazy series of threes will stretch from your screen... well, not to Saturn, of course. It won’t even reach the Sun, only a quarter of an astronomical unit, about the distance from Earth to Mars in good weather. Let me draw your attention (don’t sleep!) that a crazy tower is not a number the length from Earth to Mars, it is a tower of degrees of such height. We remember that five triplets in this tower cover the googolplex, calculating the first decimeter of triplets burns all the fuses of the planet’s computers, and the remaining millions of kilometers of degrees seem to be of no use, they simply openly mock the reader, it is useless to count them.

Now it is clear that 34 = 3333 = 337 625 597 484 987 = 3 towerless, (not 3 to the degree of towerless, but “three arrows arrow crazy”(!)), aka towerless recklessness will not fit either in length or height into the Observable Universe , and won't even fit into the supposed Multiverse.

At 35 = 33333 the words end, and at 36 = 333333 the interjections end, but you can practice if you are interested.

It is useless to calculate it, and it won’t work. The number of degrees here cannot be meaningfully counted. This number is impossible to imagine, it is impossible to describe. No finger analogies™ are applicable; there is simply nothing to compare the number with. We can say that it is huge, that it is grandiose, that it is monumental and that it looks beyond the horizon of events. That is, give it some verbal epithets. But visualization, even free and imaginative, is impossible. If with three arrows it was still possible to say something, to draw a recklessness from Earth to Mars, to somehow compare it with something, then there simply cannot be any analogies. Try to imagine a thin tower of triplets from Earth to Mars, next to another almost the same, and another, and another... An endless field of towers goes into the distance, into infinity, towers everywhere, towers everywhere. And what’s most offensive is that these towers don’t even have anything to do with the number, they only determine the height of other towers that need to be built in order to get the height of the towers, in order to get the height of the towers... so that after an unimaginable amount of time and iterations they get the number itself.

That's what g₁ is, that's what 33 is.

Have you rested? Now, from g₁, we return with renewed vigor to the assault on Graham’s number. Have you noticed how the escalation increases from arrow to arrow?

33 = 7 625 597 484 987

33 = tower, the height of the Earth to Mars.

33 = a number that is impossible to imagine or describe.

Can you imagine what kind of digital nightmare happens when the shooter turns out to be five? When are there six? Can you imagine the number when the shooter will be one hundred? If you can, let me offer you a number g₂ in which the number of these arrows turns out to be equal to g₁. Remember what g₁ is, right?

I won’t hide it, there is also g₃, which contains the g₂ shooter. By the way, is it still clear that g₃ is not g₂ “to the power” of g₂, but the number of crazy people that determine the height of crazy people that determine the height... and so on along the entire chain down to the thermal death of the Universe? This is where you can start crying.

Why cry? Because that's absolutely true. There is also a number g₄, which contains g₃ arrows between the threes. There is also g₅, there is g₆ and g₇ and g₁₇ and g₄₃...

In short, there are 64 of these g. Each previous one is numerically equal to the number of arrows in the next one. The last g₆₄ is Graham’s number, with which everything so seemingly innocently began. This is the number of dimensions of the hypercube, which will definitely be enough to correctly color the segments with red and blue colors. Maybe less, this is, so to speak, the upper limit. It is written as follows:

and they write it like this.

There was an old man, shy as a boy,
Clumsy, timid patriarch...
Who is the swordsman for the honor of nature?
Well, of course, fiery Lamarck.
Osip Mandelstam

In addition to describing Graham's number and many other interesting numbers, I'd like to discuss a couple more numbers. Now they are rushing to decipher the human genome. In my opinion, this will be of little use, like any experimental data that does not have at least some theory (it is not clear what is actually being measured). But at least it has become known that the human genome consists of 3.1 billion bases (all sorts of thymin with guanine and other uracils) Each Living being from the point of view of Darwin's theory of evolution, it is considered a test for the survival of a given combination of bases, and the main clash of religion with Darwin's theory occurs when Darwin's theory, or rather its modern interpretation, claims that this search occurs randomly. There is no contradiction outside of this statement evolutionary theory and there is no picture described, for example, in the Judeo-Christian Genesis, no matter what creationists claim there.

For example, if we assume that the very first living creature had the entire evolution from this very first creature to modern man programmed in the very first DNA, then this picture, which can be considered a modern interpretation of Lamarck’s evolution, is no different from Genesis, and the very first living the creature in this thought experiment should not be called Brodsky's Adam, but Lamarck's archetype. Simply, the words “God created” from Genesis in this context means God wrote it down in the program of the Lamarck archetype. By the way, this program and the programming method itself were also invented by Him.

Let us assume that the combination of base pairs of this very first living creature is unique, then we can estimate from below the rate of Darwin’s evolution. Let's start with the fact that the smallest living creature was recently found (viruses are supposedly even smaller, but they cannot be considered fully living creatures, since for reproduction they need someone else's cellular mechanism - all sorts of mitochondria, etc., etc.) Let's imagine that the entire universe (10 to the power of 26 meters) is filled to the brim with these living beings measuring 0.009 cubic microns who are constantly testing DNA combinations, each with their own unique test eliminating duplication of DNA testing by different living beings, and if something successful appears, then all living beings of the universe instantly learn about it and change their test task, so that all combinations based on an unsuccessful test are rejected from subsequent testing. Let's call Darwin's number the total number of genomes that need to be tested in this way, and if we multiply Darwin's number by the minimum lifetime of the testing creature - Planck time, which is the minimum quantum of time - and divide by the total number of such creatures, then we can determine a certain characteristic time of such evolution , which I propose to call Darwin's time. And if you divide Darwin's time by the maximum age of our universe, you can get a number that I propose to call William of Occam's number, since he was the first to prove that scientific methods You cannot prove the existence of God, but you cannot prove his absence either. Indeed, Occam's number shows within the framework of Darwin's theory maximum amount inputs into Darwinian evolution in our Universe, that is, it separates those DNA combinations that can be the genome of a living creature from those that are obviously fatal. That is, this number shows the difference between life and death in our Universe.

Naturally, I propose to call the ratio of the Occam number to the Graham number the Brodsky number, and I propose to call this whole procedure the Brodsky paradox.

Originally posted by lyubimica_mira at Graham Finger Number™

Original taken from sly2m in Graham Finger Number™

epigraph
If you peer into the abyss for a long time,
you can have a good time.
Mechanical Soul Engineer

Already by the first grade of school, everyone knows that there are an infinite number of numbers, they never end, and there is no such thing as the largest number. To anyone million trillion billion You can always say “plus one” and still win. And a little later the understanding comes (should come!) that long strings of numbers by themselves mean nothing. All these trillions billions They make sense only when they serve as a representation of a certain number of objects or describe a certain phenomenon. It’s not difficult to come up with a long number that doesn’t represent anything other than a set of long-sounding numbers; infinite number. Science, to some figurative extent, is engaged in looking for very specific combinations of numbers in this vast abyss, adding them to some physical phenomenon, for example, the speed of light, Avogadro’s number or Planck’s constant.

This is what a small fraction of them (about 100,000) in the Omega Centauri globular cluster looks like.

Naturally, when we are talking about such scales, the actual numbers in the number do not play a significant role, after all, everything is very conditional and approximately. May be In fact the number of stars in the Universe is “1,564,861,615,140,168,357,973,” or maybe “9,384,684,643,798,468,483,745.” Or even “3 333 333 333 333 333 333 333”, why not, although it’s unlikely, of course. In cosmology, the science of the properties of the Universe as a whole, one does not bother with such trifles. The main thing is to imagine that approximately this number consists of 22 digits, which makes it more convenient to consider it as one followed by 21 zeros, and write it as 10 21. The rule is general and very simple. Whatever figure or number stands in place of the degree (printed in small print on top of 10 here), how many zeros after the unit will be in this number, if you paint it in a simple way, with signs in a row, and not in a scientific way. Some numbers have "human names", for example we call 10 3 "thousand", 10 6 - "million", and 10 9 - "billion", but some do not. Let's say 10 59 does not have a generally accepted name. And 10 21, by the way, has it - this is a “sextillion”.

7 billion is the number of people on the planet. Based on the above, it is absolutely impossible to count them all in order during a human life; you will have to live more than two hundred years.

100 billion (10 11) - this is how many or so people have lived on the planet throughout its history. McDonald's sold 100 billion hamburgers by 1998 during its 50 years of existence. There are 100 billion stars (well, a little more) in our Milky Way galaxy, and the Sun is one of them. The observable Universe contains the same number of galaxies. There are 100 billion neurons in the human brain. And the same number of anaerobic bacteria live in the cecum of everyone reading these lines.

Trillion (10 12) is a number that is rarely used. It is impossible to count to a trillion; it will take 32 thousand years. A trillion seconds ago, people lived in caves and hunted mammoths with spears. Yes, a trillion seconds ago mammoths lived on Earth. There are approximately a trillion fish in the planet's oceans. Our neighboring Andromeda galaxy has about a trillion stars. A person is made up of 10 trillion cells. Russia's GDP in 2013 amounted to 66 trillion rubles (in 2013 rubles). From Earth to Saturn, 100 trillion centimeters and the same number of letters in total were printed in all books ever published.
Quadrillion (10 15, million billion) - that’s how many ants there are on the planet. Normal people don’t say this word out loud, well, admit it, when was the last time you heard “a quadrillion of something” in a conversation?
Quintillion (10 18, billion billion) - this is how many possible configurations exist when solving a 3x3x3 Rubik's cube. Also the number of cubic meters of water in the world's oceans.
Sextillion (10 21) - we have already encountered this number. The number of stars in the Observable Universe. The number of grains of sand in all the deserts on Earth. The number of transistors in all existing electronic devices of mankind, if Intel did not lie to us.
10 sextillion (10 22) is the number of molecules in a gram of water.
10 24 is the mass of the Earth in kilograms.
10 26 is the diameter of the Observable Universe in meters, but counting in meters is not very convenient; the generally accepted boundaries of the Observable Universe are 93 billion light years.

Science does not operate with dimensions larger than the Observable Universe. We know for sure that the Observable Universe is not the whole, the whole, the whole Universe. This is the part that we, at least theoretically, can see and observe. Or they might have seen it in the past. Or we will be able to see it someday in the distant future, remaining within the framework of modern science. From the rest of the Universe, even at the speed of light, signals will not be able to reach us, which is why these places, from our point of view, do not seem to exist. How big is that big universe In fact No one knows. Maybe a million times more than Observable. Or maybe a billion. Or maybe even endless. I’m telling you, this is no longer science, but fortune telling on coffee grounds. Scientists have some guesses, but this is more fantasy than reality.
To visualize cosmic proportions, it is useful to study this picture, expanding it to full screen.

However, even in the Observable Universe you can cram much more something other than meters.
10 51 atoms make up planet Earth.
10 80 is the approximate number of elementary particles in the Observable Universe.
10 90 is the approximate number of photons in the Observable Universe. There are almost 10 billion times more of them than elementary particles, electrons and protons.
10 100 - googol. This number doesn't mean anything physically, it's just round and pretty. The company that set itself the goal of indexing Google's links (just kidding, of course, this is more than the number of elementary particles in the Universe!) in 1998 took the name Google.
10,122 protons will be needed to fill the Observable Universe to capacity, tightly, proton to proton, end to end.
The Observable Universe occupies 10,185 Planck volumes. Our science does not know smaller quantities than the Planck volume (a cube with a Planck length of 10–35 meters). Surely, as with the Universe, there is something even smaller there, but scientists have not yet come up with sane formulas for such trifles, it’s just pure speculation.

It turns out that 10,185 or so is the largest number that, in principle, can mean something in modern science. In a science that can touch and measure. It is something that exists or could exist if it happened that we had learned everything there was to know about the Universe. The number consists of 186 digits, here it is:
100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Science, of course, does not end here, but beyond that there are free theories, guesses, and even just pseudo-scientific scratching and racing. For example, you've probably heard about the inflationary theory, according to which, perhaps, our Universe is only part of a more general Multiverse, in which these universes are like bubbles in an ocean of champagne.

Or have you heard about string theory, according to which there can be about 10,500 configurations of string vibrations, which means the same number of potential universes, each with its own laws.

The further into the forest, the less theoretical physics and science in general remains in the growing numbers, and behind the columns of zeros an increasingly pure, unclouded queen of sciences begins to appear. Mathematics is not physics, there are no restrictions and there is nothing to be ashamed of, have fun, write zeros in formulas until you drop.
I will only mention the well-known googolplex. A number that has googol digits, ten to the power of googol (10 googol), or ten to the power of ten to the power of one hundred (10 10 100).
10 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

I won't write it down in numbers. Googolplex means absolutely nothing. A person cannot imagine a googolplex of anything, it is physically impossible. To write down such a number, you will need the entire Observable Universe, if you write with a “nano-pen” directly across the vacuum, actually into the Planck cells of the cosmos. Let's convert all matter into ink and fill the Universe with just solid numbers, then we'll get a googolplex. But mathematicians (terrible people!) are just warming up with Googolprex, this is the lowest bar from which real nobodies start for them. And if you think that googolplex to the power of googolplex is what we're talking about, you have no idea HOW wrong you are.

Graham's number appeared in a paper devoted to solving one of the problems in Ramsey theory, and "Ramsey" here is not an imperfect gerund, but the surname of another mathematician, Frank Ramsey. The task, of course, is quite far-fetched from a layman’s point of view, although not very complicated, and even easily understandable.
Imagine a cube, all of whose vertices are connected by lines-segments of two colors, red or blue. Connected and colored in random order. Some people have already guessed that we will talk about a branch of mathematics called combinatorics.

Will we be able to contrive and choose a configuration of colors (and there are only two of them - red and blue) so that when coloring these segments we DO NOT end up with all the segments of the same color connecting the four vertices lying in the same plane? In this case, they do NOT represent such a figure:

I won’t even begin to explain what a four-dimensional cube is, does everyone know? A four-dimensional cube has 16 vertices. And you don’t need to rack your brain and try to imagine a four-dimensional cube. This is pure mathematics. I looked at the number of dimensions, plugged it into the formula, and got the number of vertices, edges, faces, and so on. Well, or you looked it up on Wikipedia if you don’t remember the formula. So a four-dimensional cube has 16 vertices and 120 segments connecting them. The number of coloring combinations in the four-dimensional case is much greater than in the three-dimensional case, but even here it is not very difficult to count, divide, reduce, and the like. In short, find out that in four-dimensional space you can also get creative with coloring the segments of a hypercube in such a way that all lines of the same color connecting 4 vertices will not lie in the same plane.
In the fifth dimension? And in the fifth-dimensional, where the cube is called a penteract or pentacube, it is also possible.
And in the six-dimensional.
And then there are difficulties. Graham was unable to mathematically prove that a seven-dimensional hypercube could perform such an operation. Both eight-dimensional and nine-dimensional, and so on. But this “and so on,” it turned out, does not go to infinity, but ends with some very large number, which was called the “Graham number.”
That is, there is some minimum dimension hypercube, in which the condition is violated, and it is no longer possible to avoid the combination of coloring of segments such that four points of the same color will lie in the same plane. And this minimum dimension is definitely more than six and definitely less than Graham’s number, this is the scientist’s mathematical proof.

And now the definition of what I described above in several paragraphs, in the dry and boring (but capacious) language of mathematics. There is no need to understand, but I can’t help but bring it up.
Consider an n-dimensional hypercube and connect all pairs of vertices to obtain a complete graph with 2n vertices. Let's color each edge of this graph either red or blue. For what is the smallest value of n, each such coloring necessarily contains a single-colored complete subgraph with four vertices, all of which lie in the same plane?

In 1971, Graham proved that this problem has a solution, and that this solution (the number of dimensions) lies between the number 6 and some larger number, which was later (not by the author himself) named after him. In 2008, the proof was improved, the lower bound was raised, and now the required number of dimensions lies between the number 13 and Graham’s number. Mathematicians do not sleep, work goes on, the scope narrows.
Many years have passed since the 70s, mathematical problems have been found in which numbers larger than Graham appear, but this first monster number so amazed contemporaries who understood the scale we are talking about that in 1980 it was included in the Guinness Book of Records as “the most the largest number ever involved in a rigorous mathematical proof" at that time.

Let's try to figure out how big it is. The largest number that can have any physical meaning is 10,185, and if the entire Observable Universe is filled with a seemingly endless set of tiny numbers, we get something commensurate with googolplex.

Can you imagine this vastness? Forward, backward, up, down, as far as the eye can see and as far as the Hubble telescope can see, and even as far as the Hubble telescope can, to the most distant galaxies and looking beyond them - numbers, numbers, numbers much smaller than a proton. Such a Universe, of course, will not be able to exist for long; it will immediately collapse into a black hole. Do you remember how much information can theoretically fit into the Universe? I told you.

The number is really huge, it blows your mind. It's not exactly equal to the googolplex, and it doesn't have a name, so I'll call it " dochulion". Just thought of it, why not. The number of Planck cells in the Observable Universe, and each cell contains a digit. The number contains 10,185 digits, it can be depicted as 10 10 185.
dochulion = 10 10 185
Let's open the doors of perception a little wider. Remember inflation theory? That our Universe is just one of many bubbles in the Multiverse. And if you imagine dochulion such bubbles? Let's take a number as long as all that exists and imagine a Multiverse with a similar number of universes, each of which is covered to the brim with numbers - we get dochulion dokhulion. Can you imagine this? How you float in the non-existence of a scalar field, and all around you are universes-universes and in them numbers-numbers-numbers... I hope that such a nightmare (though, why a nightmare?) will not torment (and why torment?) an overly impressionable reader at night.

For convenience, let's call this operation " flip". Such a frivolous interjection, as if they took the Universe and turned it inside out, then it was inside in numbers, but now, on the contrary, outside we have as many universes as there were numbers, and each box is full, full of numbers. Like peeling a pomegranate, You bend the crust like this, the grains turn out from the inside, and in the grains there are pomegranates again. It also came up on the fly, why not, with dochulion after all, it was a ride.
What am I getting at? Should you slow down? Come on, hoba, and one more flip! And now we have as many universes as there were numbers in the universes, the number of which was equal to up to a million numbers that filled our Universe. And immediately, without stopping, flip again. And the fourth and fifth. Tenth, thousandth. Do you keep up with your thoughts, can you still imagine the picture?

Let's not waste time on trifles, let's spread the wings of imagination, accelerate to the fullest and flip flip flips. We turn each universe inside out as many times as how many dozens of universes there were in the previous flip, which was a flip from the one before last, which... uh... well, are you following? Somewhere like this. Let our number now become, suppose, " dohuliard".
dohuliard = flip flips
We don’t stop and continue to flip dohullions of dohuliards as long as we have the strength. Until your eyes get dark, until you want to scream. Here everyone is their own brave Buratina, the safe word will be “cheese cheese”.
So here it is. What is this all about? The huge and infinite dohullions of flips and dohuliards of universes of complete digits cannot be compared to Graham's number. They don't even scrape the surface. If Graham's number is represented as a stick, stretched according to tradition throughout the entire Observable Universe, then we are here with you screwed up it will turn out to be a notch of thickness... well... how can I put it this way, to put it mildly... unworthy of mention. So, I softened it as best I could.

Now let's take a break and take a break. We read, we counted, our little eyes were tired. Let’s forget about Graham’s number, we still have a long way to go, unfocus our eyes, relax, meditate on a much smaller, even miniature number, which we’ll call g 1, and write it down in just six characters:
g 1 = 33
The number g 1 is equal to "three, four arrows, three." What does it mean? This is what a recording method called Knuth arrow notation looks like.
For details and details, you can read the article on Wikipedia, but there are formulas there, I will briefly retell it in simple words. One arrow means ordinary exponentiation.
22 = 2 2 = 4
33 = 3 3 = 27
44 = 4 4 = 256
1010 = 10 10 = 10 000 000 000

Two arrows mean, clearly, raising to a power of a power.
23 = 222 = 2 2 2 = 2 4 = 16
33 = 333 = 3 3 3 = 3 27 = 7,625,597,484,987 (more than 7 trillion)
34 = 3333 = 3 3 3 3 = 3 7 625 597 484 987 = a number with about 3 trillion digits

In short, "number arrow arrow another number" shows what the height of the powers is (mathematicians say " tower") is built from the first number. For example, 58 means a tower of eight fives and is so large that it cannot be calculated on any supercomputer, even on all computers on the planet at the same time.
5 5 5 5 5 5 5 5
Let's move on to the three arrows. If the double arrow showed the height of the tower of degrees, then the triple arrow would seem to indicate “the height of the tower of the height of the tower”? What the hell! In the case of three, we have the height of the tower the height of the tower the height of the tower (there is no such concept in mathematics, I decided to call it " crazy"). Something like this:

That is, 33 forms a crazy tower of triplets, 7 trillion high. What are 7 trillion threes stacked on top of each other and called "crazy"? If you carefully read this text and did not fall asleep at the very beginning, you probably remember that from Earth to Saturn there are 100 trillion centimeters. The three shown on the screen in the twelfth font, this one - 3 - is five millimeters high. This means that a crazy series of threes will stretch from your screen... well, not to Saturn, of course. It won’t even reach the Sun, only a quarter of an astronomical unit, about the distance from Earth to Mars in good weather. Please note (don’t sleep!) that recklessness is not a number the length from Earth to Mars, it’s tower of degrees so tall. We remember that five triplets in this tower cover the googolplex, calculating the first decimeter of triplets burns all the fuses of the planet’s computers, and the remaining millions of kilometers of degrees seem to be of no use, they simply openly mock the reader, it is useless to count them.

Let's move on to the four arrows. As you already guessed, here the crazy guy sits on the crazy guy, he drives the crazy guy around, and even with a tower, it’s the same without a tower. I’ll just silently give a picture revealing the scheme for calculating four arrows, when each subsequent number of the tower of degrees determines the height of the tower of degrees, which determines the height of the tower of degrees, which determines the height of the tower of degrees... and so on until self-oblivion.

It is useless to calculate it, and it won’t work. The number of degrees here cannot be meaningfully counted. This number is impossible to imagine, it is impossible to describe. No analogies on your fingers™ are not applicable, the number simply has nothing to compare with. We can say that it is huge, that it is grandiose, that it is monumental and that it looks beyond the horizon of events. That is, give it some verbal epithets. But visualization, even free and imaginative, is impossible. If with three arrows it was still possible to say something, to draw a recklessness from Earth to Mars, to somehow compare it with something, then there simply cannot be any analogies.
Now, from g 1, we return with renewed vigor to the assault on Graham’s number. Have you noticed how the escalation increases from arrow to arrow?
33 = 27
33 = 7 625 597 484 987
33 = tower, the height of the Earth to Mars.
33 = a number that is impossible to imagine or describe.

Everything that has been written so far, all these calculations, degrees and towers that do not fit into the multiverses of multiverses, was needed only for one thing. To show the NUMBER OF ARROWS in the number g 2. There is no need to count anything here, you can just laugh and wave your hand.
I won’t hide it, there is also g 3, which contains g 2 arrows. By the way, is it still clear that g 3 is not g 2 “to the power” of g 2, but the number of crazy people that determine the height of crazy towers that determine the height... and so on along the entire chain down to the thermal death of the Universe? This is where you can start crying.

Why cry? Because that's absolutely true. There is also the number g 4, which contains g 3 arrows between the triplets. There is also g 5, there is g 6 and g 7 and g 17 and g 43...
In short, there are 64 of these g. Each previous one is numerically equal to the number of arrows in the next one. The last g 64 is Graham’s number, with which everything so seemingly innocently began. This is the number of dimensions of the hypercube, which will definitely be enough to correctly color the segments with red and blue colors. Maybe less, this is, so to speak, the upper limit. It is written as follows:
and they write it like this:

Nobody knows what the future holds for us. Human civilization has thousands of ways to end: nuclear wars, environmental disasters, deadly pandemics, whatever asteroid might arrive, dinosaurs won’t let you lie. But nature has one unshakable law, known to us since ancient times. No matter what happens, no matter what we think to ourselves, time will not go away, it will pass. Whether we want it or not, with or without us, a thousand and 10 thousand years will pass.

What if a million years pass? But he will go wherever he goes. Graham's number, and in general everything that a person is able to think about, imagine, pull out of oblivion and make, if not tangible, but at least an entity that has some meaning, will definitely come to fruition sooner or later. Simply because today we have enough strength to develop to the ability to realize this.