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"I have this vision of hoards of shadowy numbers lurking out there in the dark, beyond the small sphere of light cast by the candle of reason. They are whispering to each other; plotting who knows what. Perhaps they don't like us very much for capturing their smaller brethren with our minds. Or perhaps they just live uniquely numberish lifestyles, out there beyond our ken."

-- Douglas Reay
            ... somewhere in the heavens ... they are waiting ... they exist in another dimension far removed from our own and our mundane concerns ... they live in the gaping yawn of eternity in the sea of infinity ... few are aware of their existence ... the contemplation of the smaller members of their kind is enough to drive men insane ... they exist in a vast realm of shadowy existence far greater than any human being can imagine ... a realm in fact far far vaster than the known universe! An elect of the smallest of its denizens are known only by arcane signs and sigils, but an infinitely greater multitude are known by no name and no sign and there is no known magic that can summon them! They make the universe tremble in freight. They are the bane of every high, noble, terrifying or transcendent concept of man ... for these shrink into obscurity in their presence and without them nothing so great or terrible can even be imagined or summoned ... what do I speak of?

                    I am speaking of the platonic realm of numbers ... a place I like to call the numberscape … a vastly unexplored realm of which only an infinitesimal (literally) fraction has ever been, or ever will be, explored and mapped. 

                    It is of course common knowledge that numbers are infinite in multitude ... that there is no end to numbers ... and that there is no largest number. Amazingly this knowledge is so common that it does not raise the slightest concern or mystery in the average person. Yet if you stop to really contemplate the implications they are indeed frightening; a runaway freight train of burgeoning mathematics unfolding unpredictably and without end until they leave the imagination quivering like jelly, unable to grasp or understand their true form ...

                    But what is so mysterious or frightening about numbers? They are by all accounts a mundane fact of life. All around the world, for at least as long as man has been civilized, practically everyone has used them all the time. They are put in the service of the most menial of tasks: the counting of household groceries, the planning of a budget, to keep score in all manner of sports, board games, computer games, etc. But they also serve as the bedrock of industry, science, and commerce for they form the basis of all physical measurement, the keeping of time, the counting of inventory ... in short our modern way of life is founded upon number.

                    They are merely a mechanical contrivance by which man goes about his business, right? Yes and No. While it is true that the signs and symbols we assign to numbers are contrivances, the numbers themselves ... the concept, not merely the forms in which they are codified, are things we recognize in the world and seek to describe through language. Did we really invent these? Did not the multitudes of the heavens exist long before any sentience could have known them, and did they not then have a number? Is some divine architect responsible for the creation of number? If so, and numbers were created, then there was a time before their existence, but then how could anything else have existed before them, for in order for anything to exist it must exist in some multitude? Even the very absence of existence is embodied in the number of zero. Then perhaps the abstract form of the multitudes are eternal ... always existent and ready to be manifested and embodied in some definite form? If so then numbers are not quite so mundane but are rather a quintessential principle of reality.

                    Yet at the same time some numbers are so common to every day experience that they are rendered quite benign. This is particularly true of the first few numbers, such as one,two, and three which by an odd inverse law are the rarest of numbers but the most frequently used. Every school child knows how to count every number from one to one hundred. In the modern world we are of course saturated with numbers much larger than this. We hear of millionsbillions, and trillions with such frequency that they have become household words, lost of their significance. How often have you heard people use million hyperbolically? The result is that a million doesn’t seem like all that much. This is why descriptions of a million often come across as surprising and counter-intuitive. Our perception of numbers is very much distorted by our very limited ability to grasp them. To make matters worse we often hear of even larger numbers expressed in scientific notation in popular science discussions. The result is a general sense that large numbers are not that large, and don't get that big. But the compactness of scientific notation already belies terrifying realities … realities that make trillions tremble in fear. Far from being mundane we see that the size of the cosmos is something frightful to human dimensions … it’s something we can’t quite wrap our brains around no matter how hard we try. We are forced to admit to ourselves our own abysmal ignorance and impotence before thrones and dominions, powers and principalities. And it is here that the frightful implications of numbers comes into play … for you see the reality you know is but a pale pin point of light in a far vaster … and in fact a far more real reality of shadow ... in fact a hierarchy of shadows veiled within shadows. The universe is a great analogy here … for it is indeed almost entirely void, and our earth but a pale blue dot within it. Our reality depends on the greater reality but the greater reality would barely notice if we were gone. But this horror pales in comparison to the horrors that must be so if ' numbers are without end ', for this implies that the universe itself is but an infinitesimal dot in an infinite sea of unknowing, unknown, and unknowable! If ' numbers are infinite ', but are also in some sense real then we really know virtually nothing about them. We may speak of millions, billions, and trillions … but where we know not … we can speak not. This shadowy “out there” is of course implicitly understood even by the laymen of mathematics. The mathematician himself easily glides over this fact with a simple ellipsis ( … ). But if scientific notation belied horrors of great magnitude, the ellipsis hides incomprehensible horrors beyond incomprehensibility!

            It therefore is not the numbers themselves which are mundane, but only the earthly guise they assume to be in some way tangible to us. That is to say, our conception of them. The numbers that are common place to us are precisely those which are most useful, most immediate, and most easily expressed. But I speak of numbers beyond this common fray of daily experience ... numbers that escape our grasp to express them in common language and yet are no less real multitudes than those we easily grasp. These are the terrors of which I wish to contemplate.

                        But WAIT, What’s all this talk of horrors, incomprehensibility, and the unknown? Surely we know the numbers, at least in principle. Sure they are infinite in number, but they can be understood inductively. First we have one, then we form two by finding it's successor, then we form three by finding it's successor, as so on ad infinitum. Thus we have constructed the set of counting numbers by virtue of the fact that we have described anything that we might call a counting number. Therefore we know the counting numbers. But is describing what a counting number is the same as describing all the counting numbers? All we have done is provided a definition which characterizes a counting number. Having a definition of a certain category is not the same as knowing the content of that category. It only means that we could, in principle, recognize something belonging to that category if we saw it. But we wouldn't know what it's contents were in advance.

                    To illustrate, we might know what a cat is, and isn't, but that doesn't imply we know all cats. We may not know any cats at all. To know a particular cat would mean to know something about it besides the fact that it was a cat, such as it's fur pattern, eye color, temperament, etc. Not every combination of traits would necessarily be represented by the members of this collection and we wouldn't know in advance which were and which weren't. Knowing what a cat is tells us nothing about how many cats there are or how the various cat-like traits are distributed amongst them.

                    Likewise, the counting numbers can be said to have inherent traits that characterize them, such as being even, odd, prime, perfect, square, etc. Likewise not every combination of traits would necessarily be represented by the members of the collection. For example there are no square primes. Some combinations of traits we may not even be able to say exists or not without at least one example, or a proof of their non-existence. Such is the case for odd perfects. Furthermore the distribution pattern of all number-theoretic traits is not something that either the definition of the counting numbers, nor the traits themselves, automatically provides us with. It only enables us to identify which numbers have which traits, or may suggest ways of either finding numbers with those traits or proving their non-existence. Consider this: if we know all counting numbers why don't we, for example,  know whether or not there are an infinite number of twin primes?

                    Because we don't know all the counting numbers, in the same way that we know say, one,two, and three! These numbers are known. Humanity in some sense has lay claim to these numbers by virtue of the fact that we have named them. Numbers do not inherently have names any more than trees, rocks, stars, etc. We give things names to make them more tangible and recognizable to us. This is especially true of abstract concepts.

                    The first three counting numbers are further known to us in that we know some of their properties. We know that one is odd, and that two and three are primes. They are familiar to us by virtue of the fact that they have been concretely realized and represented an innumerable number of times throughout human history. The popularity of these three numbers, in fact, far out classes the popularity of any other numbers. Sages have spoken their names in hallowed tones, put them on a pedestal as sacred truths and associated them with all manner of magic, philosophy, theology, and mysticism. Granted there is something rather special about them being at the very beginning of the sequence ... but none the less three is a very tame little creature that sits obediently in the palm of your hand. It's a number so small that the mind of man can fully comprehend it. You can see three immediately and without the need to count it out! Not only are the first three numbers found everywhere in the domain of man, but they also occur everywhere in nature. The one tree, the branch forking into two paths, and the three-leafed clover, are just a few examples. These numbers are so easily realized, so frequently represented, that they are indeed a banality of reality.

                    But I did not come here to simply speak of the commonly known, nor to just wax philosophical about the unknown. Instead I want to speak of a world well above the fray of the everyday and ordinary.

                    The numbers one, two, and three are numbers everyone knows. Tens, hundreds, thousands, millions, billions, and trillions are the kinds of numbers most people know. But there is a world of much much larger numbers out there that most people don't know and would never think to imagine. Numbers beyond the everyday, the commonplace, but also beyond the astronomical, cosmic, and beyond the known universe. There are numbers lurking out there ... huge ones ... mathematical behemoths ... of such complexity that the universe would pass away long before computing them! These numbers are not rare ... in fact they far outnumber the little piddling numbers we use to measure the national debt, or even count the stars or atoms of the known universe. They outnumber them to such an extent that our familiar numbers vanish into a infinitesimal point in a world composed almost entirely of darkness. These numbers exist in a realm so rarely visited by the thoughts of man that the silence and loneliness is deafening. No one, and nothing can hear you scream out there. Yet at the same time the place is so full of eldritch abominations that they are literally everywhere packed right up against each other!

                    The few men and women who have dared to travel to these far off places of the mind have only managed to leave a very few markers along a path away from the familiar and towards a great and terrible infinite unknown. Yet no man can transverse even the marked path fully. One must travel by leaps and bounds. Counting in succession would not get you to these far far places! Not in a life time, not even in a thousand life times. One must teleport there by means of mathematical sling shots. What one passes on ones way to their destination however are vast fields of numbers which have never been and may never be known.

                    The reason for this is something you would never suspect from the definition of the counting numbers: that the playing field of numbers is not level, but is in fact riddled with unfathomable peaks and abysmal valleys. The peaks represent numbers we can easily express, while the valleys represent numbers we can never express. In order to express any number we need a description of it. Naively we might assume that the length of the description of a number is directly proportional to its size. If this were true we'd never be able to speak of anything beyond the cosmic, or even the terrestrial. While there is a general trend towards longer descriptions as the size of the numbers increases, we find that the length of the description does in fact vary wildly even for numbers of the same ' weight class '. An extremely large number may have a description which can be expressed in just a few characters, while a slightly smaller but still extremely large number may have a description no shorter than the number itself. As a general rule, for any given range, the numbers with descriptions no shorter than the number itself will tend to vastly outnumber the numbers which have descriptions short enough for us to actually describe! Put another way, given any sufficiently large number, it will not be possible to express every number less than it no matter how hard we try.

                Something like this can not be said, even of seemingly very large numbers like those discussed in science. Every number in mainstream science could be described with less than a hundred digits, and therefore any one of those numbers at any given time can be described easily by simply writing out its decimal form.

                The only clear trend we discover is that, the least amount of information required to describe a number above a certain size, gradually increases with size. However even this increase is not steady and predictable. It strongly resists increase even as we forge on ahead to numbers that would make the heads of numbers just moments ago spin, and even when we think we see a trend where it's steadily increasing it suddenly surprises us with a new innovation that keeps the information from increasing too rapidly. Consequently we find that the distribution of peaks is highly irregular and unpredictable.

                    The complexity and vicissitudes of the numberscape could hardly be guessed, let alone known, from the mere definition of the counting numbers and successorship. This is a direct consequence of the fact that numbers, although seemingly possessing merely a sequential relation to each other, in fact contain structure. This structure is usually invisible to the untrained eye, but with mathematics we can actually begin to see these structures. The more structured a number is the more self-similarities it has, and consequently, the more compact it's description can become. The fact of primality is a directly related to this, for primes are numbers which try to avoid self-similarities, where as composites possess self-similarity because they are whole multiples of other numbers. The order and disorder of which I speak of is just a large scale version of the concept of primality. As such it exhibits many of the same basic properties: a general trend which is marred by an unpredictable randomness, and an impossibility of a complete description of its overall design.

                    The numberscape has attributes similar to those of fractals. Like fractals the numberscape begins with a very simple definition which leads to infinite complexity. Like fractals numbers will contain features of self-similarity, and resemble super-sets of smaller numbers. Lastly, like fractals it contains much that appears random and chaotic, even while being completely deterministic. In short, we discover that even the simple counting numbers have infinite mysteries for mathematicians to solve. 

                    The unknown beckons us onwards beyond the familiar world and into one of pure abstraction.

                    Unbeknownst to most people, human beings have ventured far far beyond numbers they would ever have any practical use for. When most people think of large numbers they think of a number with a huge number of digits. They are unaware that man has gotten so far beyond this that to ask "how many digits" certain very large numbers have has become laughably naive. Even the number of digits becomes something that can not be explained in ordinary terms. These numbers can only be described in the most arcane and esoteric of mathematical language. Mathematics has enabled us to create a vast system of numbers so far beyond any other human endeavor that any comparison to anything else becomes absolutely absurd.

                    On the subject of coming up with large numbers my brother once commented "couldn't you just keep adding zeroes". My response was "yes you could, but anyone can do that". His response was "that's my point", to which I retorted "that's my point. It can't be something obvious, it has to be something that your competitor wouldn't think of". My brother, apparently still not convinced then said "well what about something like one followed by a million zeroes". I shook my head and informed him it wasn't nearly big enough. Next he offered "one followed by a million zeroes raised to the power of one followed by a million zeroes". Again I shook my head. Then he suddenly blurted out "well how big do these numbers need to be?!". I don't remember my response, but I believe the best answer to this is: as big as we can imagine them to be ... need has nothing to do with it ...

                    Well if we want something as big we can possibly imagine, why not just go with infinity then? Infinity is bigger than any other number so your guaranteed to trump any other number, right? Well first off, what if someone else already came up with infinity (hint: very likely), and what about infinity and one? If infinity is okay why not infinity and one? But then we'd just be back where we started in a stalemate so there wasn't much point in introducing infinity as a game-breaker in the first place!

                    ... And that's really what infinity is, a game-breaker. It's an attempt to forestall the unending quest for larger and larger numbers by concocting a largest possible number. Infinity seems like a good candidate for this since it seems to be, by definition, larger than any other number, and adding to it doesn't seem to make it any larger. But recall that in the beginning it was already stated as common knowledge that there is no largest number!

                    A relatively good case can be made that infinity is not a number at all. It's very name hints at this. There is a reason why infinity has been described as immeasurable and without number. It's not just that infinity can't be measured or numbered in practice ... it can't even be measured or numbered in principle! Infinity just means unbounded, but the numbers themselves are already unbounded. In order to define infinity we must exhaust the list of numbers, an absurd and contradictory task to begin with! But assuming that we can speak of the whole of all numbers, or the emptying of all numbers, what number is left that we might call infinity? If infinity is a number it should be included in the list of all numbers. But then it would have been removed at some stage of the emptying, and thereby could not represent the totality of all numbers. Therefore there is no number that measures or numbers the infinite, because they are all (not some, but ALL) used up before we "reach" infinity (which is absurd because you can't exhaust an inexhaustible process, nor can you reach the end of an endless process). There is no number which can be representative of all numbers ... there is no number for the number of numbers ... there is no largest number! Thus we conclude that infinity is NOT a number! Rather it is an expression of the exhaustible nature of numbers.

                    Some will argue however that all this proves is that infinity is not a counting number. However I'd like to point out that this is only because we have made a special distinction between finite and infinite numbers. If we simply speak of numbers in the general sense, and then speak of the whole of all numbers also being a number we run into the same paradox. 

                    But perhaps this might be considered splitting philosophical hairs. So I'll grant for the moment that infinity is a number, it's just in a different class of numbers than the counting numbers. When I speak of coming up with the largest possible numbers then, I mean in the stricter sense of counting numbers, numbers which are actually bounded measures.

                    But if our goal is big, bigger, biggest (well not so much that last one) then why should infinity be excluded? 

                    Because summoning the unbounded is not really a challenge. The challenge is to find a bounded number that is as large as possible. You may think that if we can summon the unbounded (infinity) and we can summon one both with equal ease, then it must follow it is equally easy to summon any number within that interval. It is not. Try to resist the temptation to invoke infinity, and try to come up with the largest finite number you can. Chances are, if you are new to this topic ... your number is too small. It's no where near the kinds of numbers I'm speaking of. Although it may feel like you can imagine any number, in fact your imagination is itself bounded. You must first expand your imagination before you can have a chance of even remotely contemplating these kinds of numbers.

                    So in a nutshell we won't speak much about infinity because infinity is known. The definition of infinity, as a non-terminating process, just like the definition of the counting numbers, or the meaning of the numbers one,two, and three is known. In that sense it's uninteresting. What is not known are all the numbers that lie between one and infinity! Hence the title of this book. It is these numbers which will be the main subject of this book. 

                    Infinity is also a conversation killer. It usually signifies the end of a conversation about large numbers, not the beginning. It is because of this very reason that I'm addressing infinity at the outset. By leaving the discussion at infinity we miss out on all the amazing things to learn about finite numbers. Remember that as a whole they are anything but finite! In fact I believe that studying the finite is the most sincere way to appreciate the infinite. People who have studied large numbers have often said that the numbers were so large that they were literally bigger than their concept of infinity, and their mind had to expand to accommodate them. The fact is we can't imagine the infinite and all our concepts of it are essentially illusions. We simply imagine a size big enough such that the boundary is blurred. This is why people get a sense of vastness when contemplating infinity ... because we are just imagining some large subset of it. Rather than describe infinity as vast, it would be more accurate to describe it as vastless, in the same sense that it is immeasurable and innumerable. To have any contemplation of vastness we need to consider some sort of boundary by which to judge it's vastness relative to ourselves, but infinity is without boundary. What's interesting is that our sense of the scale of the infinite can shrink and grow. Since the largest numbers most people have any sense of are astronomical , they tend to think of the infinite as astronomically large. Perhaps that even sounds accurate to you. But the astronomical does not capture the unfathomable vastness of the least of the denizens who reside within the infinite. Thus by studying large finite numbers you will expand your very idea of infinity and your concept of it will never be the same. THAT is why to speak of infinity is to defile it, and to speak of the finite is to glorify it! Before you envoke the infinite, envoke the finite and see how your terror grows...

                    But for some, the temptation to invoke infinity will be too great. But let me explain why such an impulse is ultimately lazy. Usually this is done in the vein of one-upmanship. Someone has come up with a truly vast finite number. If you can not understand it then you have no real way to out perform them. Simply "adding one" will look cheap, since you didn't come up with the original number yourself. So how to you trunce them: you invoke infinity of course! That's guaranteed to be bigger! But rather than being your moment of triumph you have really just admitted your abysmal ignorance. Since you couldn't legitimately compete you cheated by taking the easy way out. In essence you have learned nothing and proved nothing. Literally ANYONE can invoke the infinite. But NO ONE can invoke all of the finite. It's a curious counter-intuitive fact.

                    It turns out however, that even if we admit infinity however, the "game" doesn't end. Rather than having reaching the pinnacle of one-upmanship you will quickly find that now anything is fair game. Infinity and one, two infinity , infinity squared, infinity raised to infinity , and so on. In essence this game plays out exactly like that for finite numbers, because we are simply treating infinity as an unspecified large quantity with all the prerequisite properties of numbers. But if we are going to do that anyway, why not just replace infinity with a specific large quantity. This gives us the advantage of having a definite quantity rather than an indefinite one. It's not as if we lose anything ... we can still continue the game indefinitely. It's not as if the finite numbers are going to run out!

                    A problem with the above hierarchy of infinities however is that it's not entirely clear that we are actually getting anywhere. Infinity is not just another large number. It has properties radically different from the counting numbers. One can prove that all the above infinities are in fact of the same numerosity as plain old vanilla infinity. That is, we can prove that infinity and one, two infinity, infinity squared, infinity raised to infinity, and so on, are all equal to infinity!

                    But even if we somehow ignore that, the very idea that we could have such a sequence where these represent different sizes of infinity we run into a serious problem. For we define infinity as the first quantity after all finite quantities. However if infinity is the first such quantity, then is infinity minus one infinite or not? If it's finite then infinity isn't infinite, but if it's infinite, then infinity isn't the first infinite number! As you see treating infinity as a number, or at least as an ordinary number, only leads to paradoxes.

                    We can get around this problem if we instead define infinity as the measure of the whole of the counting numbers. In this case, adding one doesn't make it any bigger, and taking one away doesn't make it any smaller. Just consider, if we had all the numbers, and we added the letter "a" to the collection, is the collection any more numerous, or is it still infinite. Likewise, if we removed one from the collection of all numbers, wouldn't we still have an infinite number of numbers left? But here we run into even a bigger problem, for it turns out that under this definition there are even larger infinities which we can not say is equal to the smallest infinity. In fact it can be proven that it is impossible to demonstrate them to be equal. So even then the numbers don't end and we don't get an ultimate trump card. There is always a bigger trump card it seems. 

                    In that case why not speak of these infinities. The reason is because it is not even entirely clear that the second smallest known infinity is something that really makes sense. If it is real, it's a kind of infinity we can't even construct in principle if we had an unlimited amount of time and resources. This can not be said of ordinary infinity. We may not be able to actually list out all the counting numbers but we can at least define them. We also could at least summon any particular member in principle, as long as we had a sufficiently large (but still finite) amount of time. In other words all the members of an ordinarily infinite collection can be constructed at least in principle. However for a collection of this higher order infinity, the vast majority of it's members can not, by necessity, be constructed even in principle. In fact only an ordinary infinity of members of this collection can be constructed in principle! How then can we even say that the members of such a collection exist, if we can't even give one example of them? And if it's members don't exist how can such a greater totality exist?!

                    But I'll even grant you that these greater infinities exist and all their members exist ... they just are things which can never be known to mortal man. Fine. But we still have a problem. For you see the second smallest known infinity, is still not the largest infinity, and we if we allow one new kind of infinity we get an infinity of new infinities ... in fact even that is an inadequate expression now! The infinities just keep on going ... quickly spinning out of control ... and with no guarantee that we're even speaking sense. Eventually the dam breaks and sheer madness ensues. At some point it isn't even clear what kinds of infinities are real and which are just the products of a fevered imagination. To make matters worse, it becomes impossible to compare such infinities that result from different theoretical continuations of infinity...

                    In short, if we open the flood gates to infinity what we get is not that the game ends, but that it continues at a totality other level ... but this time nothing is as clear and well defined. Unlike infinity, no matter how large the finite numbers get we can be sure that they maintain some sort of platonic existence. After all, they can all be constructed in principle with a sufficient amount of time. Thus I find the game with large finite numbers to be a largely more fun version of a similar game played with large infinities. As it turns out we really don't have to make much of a trade off because, as you'll learn if you continue, there are ways to bring the infinite in the service of the finite. So we get a smattering of the theory of higher infinities to boot!

                    We will visit the subject of infinity again at the end of the book, but before that I want to welcome you to the wonders of the numberscape. I guarantee you it's unlike anything you've ever imagined before ...

The Call to Adventure...


                    Large numbers have inspired both fear and awe in the minds of men since their dawn. In ancient times they were the domain of the cosmos and the divine. Today they are treated as little more than a mathematical curiosity; fit for attracting the layman to mathematics and little else. None the less, large numbers have plagued the imaginations of many a professional and amateur alike. They are ominous, austere, vast beyond description, godly, transcendent, and often difficult or impossible to fully grasp. They are both a celebration of the human spirit, and an affront to it; For while we may revel in their vastness, get lost amongst their mindless yet tireless machinations, we are also forced to confront our potential insignificance in a reality that stretches beyond all human comprehension!

            The study of large numbers can be a deeply spiritual experience, for it taps into the very essence of what it means to leave the profane behind in pursuit of the transcendent. While it is by no means a practical Endeavor, most of these numbers leaving our picayune existence far behind, it is at its core an expression of the unquenchable human spirit, which always yearns for more. It is our attempt to grasp what is beyond ourselves; to understand what we may call the divine.

        I entreat you therefore, if my words have had any impact upon you, to join me on a quest through the infinite multitudes of the finite. A word of caution before we proceed: the traveling will only get more difficult as we proceed, and with no end in sight. The venues large numbers can open up in your mind can be quite rewarding, but be warned that madness lies this way. For we can no more imagine the end of numbers than we could wait out eternity. An earnest study of large numbers will completely shift the way you think about infinity, and the finite, and your mind shall be forever changed by it.

        If these warnings have not turned you away, and you are still eager to learn the deep secrets of the numberscape, then I invite you to read onwards, as casually or religiously as you’d like. Large Numbers are not the domain of any one man, or group, for there is a number for every man that ever lived, ever will live, or ever could live; ample real estate for all humanity for all ages amongst the greatest of all heavens! What lengths are you prepared to go to try and reach infinity, even as you know the task is impossible? Why not see how "close" you can get… ready ... one,two,three,... go ...

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