"Skewered!" (1974)
I don't write many mathematical articles in this series, and for a very good reason. I don't have a mathemati- cal mind and I am not one of those who, by mere thought, finds himself illuminated by a mathematical concept.
I have, however, a nephew, Daniel Asimov by name, who does have a mathematical mind. He is the other Ph.D. in the family and he is now an Assistant Professor of Mathematics at the University of Minne- sota.
Some years ago, when he was yet a student at M.I.T., Danny had occasion to write to Martin Gard- ner and point out a small error in Gardner's excellent "Mathematical Recreations" column in Scientific American. Gardner acknowledged the error and wrote me to tell me about it and to ask a natural question. "Am I correct in assuming," said he, "that Daniel Asi- mov is your son?"
Well! As everyone who knows me knows, I am only a little past thirty right now and was only a little past thirty at the time, some years ago, when this was tak- ing place. I therefore wrote a letter to Gardner and told him, with some stiff ness: "I am not old enough,
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Martin, to have a son who is old enough to be going to M.I.T. Dannv is the son of mv vounger brother."
Friends of mine who have heard me tell this story keep assuring me that my statement involves a logical contradiction, hut, as I sav, I do not have a mathemat- ical mind, and I just don't see that.
And yet I must write another mathematical article now because over eleven vears ago I wrote one in which I mentioned Skewe-s* number as the largest fi- nite number that ever showed up in a mathematical proof.w Ever since then, people have been asking me to write an article on Skewes" number. The first re- quest came on September 3, 1963, almost immediately after the article appeared. On that date, Mr. R. P. Boas of Evanston, Illinois, wrote me a long and fasci- nating letter on Skewes' number, with the clear inten- tion of helping me write such an article.
I resisted that, along with repeated nudges from others in the vears that followed, until March 3, 1974, when, at Boskone 11 (a Boston science fiction con- vention at which I was guest of honor), I was cor- nered by a fan and had Skewes' number requested of me- So I gave in. Eleven years of chivvying is enough.! I am Skewered.
First, what is Skewes' number? Not the numerical
- See "T-Formation," reprinted in Adding a Dimension (New York: Doubleday, 1964).
f 111 admit that I've been chivvied longer than that in some respects. For seventeen years I have been requested, with varying degrees of impatience, to write another U)e Baley novel; and for over twenty years to write another Foundation novel. So please don't anybody write letters that begin with "If eleven years of chivvying is enough, why don't you——." Because I'm doing all I can, that's why.
expression, but the significance. Here's the story as I got it from Mr. Boas (though I will paraphrase it, and if I get anything wrong, it's my fault, not his).
It involves prime numbers, which are those num- bers .that cannot be divided evenly by any number other than themselves and one. The numbers 7 and 13 are examples.
There are an infinite number of prime numbers, but as one goes up the list of numbers, the fraction of these numbers that are prime decreases. There is a formula that tells you the number of primes to be found in the list of numbers up to a given number, but like everything else about prime numbers, the for- mula is not neat and definite. It only tells you approx- imately how many primes there will be up to some limiting number.
Up to the highest limit that has actually been tested, it turns out that the actual number of primes that exist is somewhat less than is predicted by the formula.
In 1914, however, the British mathematician John Edensor Littlewood demonstrated that if one length- ened the string of numbers one investigated for primes, one would find that up to some limits there would indeed be less than the formula predicted, but that up to other limits there would be more than the formula predicted.
In fact. if one continued up the line of numbers for- ever, the actual total number of primes would switch from less than the formula prediction to more than the formula prediction to less than the formula prediction, and so on—and make the switch an infinite number of times. If that were not so, Littlewood demonstrated, there would be a contradiction in the mathematical structure and that, of course, cannot be allowed.
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The only trouble is that as far as we have actually gone in the list of numbers, not even one shift has taken place. The number of primes is always less than the formula would indicate. Of course, mathemati- cians might just go higher and higher up the list of numbers to see what happens, but that isn't so easy. The higher one goes, the longer it takes to test num- bers for primehood.
However, it might be possible to do some theoreti- cal work and determine some number below which the first switch from less than the prediction to more than the prediction must take place- That will at least set a limit to the work required.
Littlewood set S. Skewes (pronounced in two sylla- bles, by the way, Skew'ease) the task of finding that number. Skewes found that number and it proved to be enormously large; larger than any other number that ever turned up in the course of a mathematical proof up to that time, and it is this number that is popularly known as "Skewes' number."
Mind you, the proof does not indicate that one must reach Skewes' number before the number of primes shifts from less than the prediction to more. The proof merely says that some time before that number is reached—perhaps long long before—the shift must have occurred.
A number as large as Skewes' number is difficult to write. Some shorthand device must be used and the device used is the excellent one of exponential nota- tion.
Thus, 1000 = 10 X 10 X 10, so 1000 can be written as 10s (ten to the third power), where the little 3 is called an "exponent." The little 3 signifies that 1000 can be considered the product of three 10s, or that it can be written as 1 followed by three zeros. In gen-
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eral, 10s (ten to the xth power) is the product of x 10s and can be written as a 1 followed bv x zeros.
Since 10,000,000,000 is written as a 1 followed by 10 zeros, it can be written exponentially as 101" (ten to the tenth power). In the same way, a 1 followed by ten billion zeros, something that would be imprac- tical to write, can be expressed exponentially as lO10-"00-000-"'10 (ten to the ten billionth power). But since ten billion is itself 1010, lo10.000.000.0"0 can be written, even more briefly, as 101010.
Writing exponentials is always a strain when an arti- cle is being written for a nonspecialized outlet- This is especially so when one is forced to place exponents on exponents. To avoid driving the Noble Printer crazy and to make the notation look prettier, I have in- vented a notation of my own. I make the exponent a figure of normal size and it is as though it is being held up by a lever, and its added weight when its size grows bends the lever down. Thus, instead of writing ten to the third power as 103, I will write it as 10\3.
In the same way, ten to the ten billionth power can be written as 10\ 10,000,000,000 or as l0\10^10.
Using this "Asimovian exponential notation," Skewes' number becomes 10\10,10\34.
Now let's see what Skewes' number might be in or- dinary nonexponential notation. To do that, we must consider the components of the exponential notation from right to left. Starting at the right, we know what 34 is, we move leftward and consider 10'\34. This is ten to the thirty-fourth power and can be written as a 1 followed by 34 zeros thus: 10,000,000,000,000,000,000,- 000,000,000.000,000, or, in words, ten decillion (Ameri- can style). This means that Skewes' number can
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be written ten 10\10\10,000,000,000,000.000,000.000,- 000,000,000,000.
So far, so good, if a bit disconcertingly formidable. The next step is to move one place to the left and ask how we might write: 10\ 10.000,000,000,000.000,000,- 000,000,000,000,000. Easy. You just put down a 1 and then follow it by ten million billion billion billion (or ten decillion, if you prefer) zeros.
If vou were to try to write such a number by begin- ning with a 1 and then writing ten decillion zeros, each the size of a hydrogen atom, you would require nearly exactly the entire surface of the Earth to write the number. Furthermore, if you wrote each zero in a trillionth of a second and kept it up at that rate with- out cessation, it would take a thousand trillion years to write the entire number.
Anyway, let's call this number the "Earth-number," because it takes the Earth as a blackboard to write it. and imagine that we can write it. Now we can write Skewes' number as 10\ Earth-number, and this means we now know how to write Skewes' number in the usual fashion. We start with a one and then follow it with an Earth-number of zeros.
This is tremendously more than the ten decillion ze- ros it took merely to write the Earth-number. A num- ber itself is much greater than the number of zeros it takes to write it. It takes only one zero to write 10, but the result is a number that is ten times greater than the number ^>f zeros required to write it. In the same way it takes ten zeros to write 10,000,000,000, but the number written is ten billion, which is a billion times greater in size than the number of zeros used to write it.
Similarly it takes only ten decillion zeros to write
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the Earth-number, but the Earth-number itself is enormously greater than that number or zeros.
To write not ten decillion zeros, but an Earth- number of zeros, would require far more than the sur- faces of all the objects in the known universe, even with each zero the size of a hydrogen atom. A trillion such universes as ours might suffice, and that is just to write the Earth-number in a one followed by zeros. Skewes' number itself, written by a one followed by an Earth-number of zeros, is enormousiy, ENORMOUSLY greater than the Earth-number tliat suffices to count those zeros.
So let's forget about counting zeros; that will get us nowhere. And if we abandon counting zeros, we don't need to have our exponents as integers. Every number can be expressed as a power of ten if we allow deci- mal exponents. For instance, by using a logarithm ta- ble, we can see that 34=10\1.53. So instead of writ- ing Skewes' number as 10^10^10^34, we can write it as 10\10'\10\10\1.53. (Such fractional expo- nents are almost always only approximate, however.)
There are some advantages to stretching out the large numbers into as many tens as is required to make the rightmost number fall below ten. Then we can speak of a "single-ten number," a "double-ten number," a "triple-ten number," and so on. Skewes' number is a "quadrupie-ten number."
We can't count objects and reach Skewes' number in any visualizable way. Counting zeros is no help either. Let us instead try to count permutations and combi- nations,
Let me give you an example. In the ordinary deck of cards used to play bridge, there are fifty-two dif- ferent cards. (The number 52 is itself a single-ten
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number, as are all the numbers between 10 and 10,000,000,000; 52^10\1.716.)
In the game of bridge, each of four people is dealt thirteen cards. A player can, with equal probability, get any combination of thirteen cards, and tlie order in which he gets them doesn't matter. He rearranges that order to suit himself. The total number of differ- ent hands lie can get by receiving anv thirteen cards out of tlie fifty-two (and I won't bother vou with how it i.s calculated) is about 635.000,000,000. Since this number is higher than ten billion, we can be sure it is beyond tlie single-ten-number stage. Exponentially, it can lie expressed as 6.35X10\11. Logarithms can help us remove that multiplier and put its value into the exponent at the cost of making that exponent a decimal. Thus 6.35 X10.11= 10\11.80. Since 11.80 is over ten, we can express that, exponentially, as 11.80 = 10'\1.07.
Consequently, we can sav that the total number of different hands a single bridge plaver can hold is 10'\10\J..07. Using onlv thirteen cards, we have, in a perfectly understandable way, reached a double-ten number. We might almost feel that we were halfway to the quadruple-ten number that is Skewes*.
So let's take all fifty-two cards and let's arrange to have the order count as well as the nature of the cards. You begin with a deck in which tlie cards are-in a certain order. You shuffle it and end witli a differ- ent order. You shuffle it again and end with yet an- other order. How many different orders are there? Remember that any difference in order, however small, makes a different order. If two orders arc iden- tical except for the interchange of two adjacent cards, they are two different orders.
To answer that Question, we figure that the first
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card can be any of the fiftv-two, the second any of the remaining fifty-one, the third any of the remaining fifty, and so on. The total number of different orders is52X51X50X . . . 4X3X2X1, In other words, the number of different orders is equal to the product of the first fifty-two numbers. This is called "factorial fifty-two" and can be written "521"
The value of 52! is, roughly, a one followed by sixty-eight zeros; in other words, a hundred decillion deciltion. (You are welcome to work out the multipli- cation if you doubt this, but if you try, please be pre- pared for a long haul.) This is an absolutely terrific number to get out of one ordinary deck of cards that most of us use constantly without any feeling of being overwhelmed. The number of different orders into which that ordinary deck can be placed is about ten times as great as all the subatomic particles in our en- tire Milky Way galaxy.
It would certainly seem that'if making use of thir- teen cards with order indifferent lifted us high up. making use of all fifty-two and letting order count "will do much better still—until we try our exponential notation. The number of orders into which fifty-two different cards can be placed is 10\68 == 10\10\1.83.
That may strike you as strange. The number of or- ders of fifty-two cards is something like a trillion tril- lion decillion times higher tlian the number of bridge hands of thirteen cards; yet, while the latter is 10\10\1.07, the former is only 10\10\1.83. Were still in the "double-ten numbers" and we haven't even moved up much,
The trouble is that the more tens we add to such exponential numbers, the harder it is to move that rightmost component. For instance, a trillion is ten times as great as a hundred billion, and counting a
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trillion objects would be an enormously greater task than counting a hundred billion. Write tliem exponen- tially, however, and it is 10.12 as compared with 10\11, and the rightmost components are only a unit apart. Write the twelve and the eleven as powers of ten so that vou can make use of double-ten numbers, and a trillion becomes 10\10\1.08, while a hundred billion is 10\10\1.04 and the difference is scarcely noticeable.
Or put it another way. The number'10\3 (which is 1000) is ten times as high as 10\2 (which is 100), but the degree to which 10\10\3 is greater than 10\10\2 would require a 1 followed by 900 zeros to be expressed. As for comparing -10\10\10\3 and 10\10\10\2, I leave that to you.
This is disheartening. Perhaps reaching the quadruple-ten numbers won't be that easy after all.
Let's try one more trick with fifty-two cards. Sup- pose each of the cards can be any card at all. Suppose the deck can have two tens of diamonds or three aces of clubs, or, for that matter, fifty-two threes of hearts. The total number of orders of such a chameleonic deck could be calculated by imagining that the first card could be any one of fifty-two, and the second card could be any one of fifty-two, and so on for all fifty-two. To calculate the number of different orders, you would have to take the product of 52 X 52 X 52 X ... 52 X 52 X 52; fifty-two 52s. This product which could be written 52\52 I might call "superfactorial fifty-two," but if so, I would be using a term I have just made up, so don't blame the mathe- maticians.
Superfactorials are immensely larger than factorials. Factorial fifty-two can be expressed by a one fol- lowed by sixty-eight zeros, but superfactorial fifty-
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two is one followed by ninety zeros—ten billion tril- lion times higher. Yet express it exponentially and superfactorial 52 == 10\90 = 10\10'\1.95.
No good. We're still in the double-ten numbers.
Well just have to forget playing cards. We must have more than fifty-two units to play with, and we had better go all the way up; all the way up.
A generation or so ago, the British astronomer Ar- thur S, Eddington calculated that the total number of electrons, protons, and neutrons in the universe was 10\79, or 10\10\1.90. This number is arrived at if we suppose that the sun is an average star, that there are about a hundred billion stars in the average gal- axy, and that there are a hundred billion galaxies in the universe.
In addition to electrons, protons, and neutrons, of course, there are numbers of unstable particles un- known to Eddington, but their numbers are compara- tively few. There are, liowever, massless particles such as neutrinos, photons, and gravitons, which do not gen- erally behave like particles but which are very numer- ous in the universe.
If we wisli, we can suppose that the number of massless particles speeding through space at any time is nine times the number of massed particles (proba- bly a grievous overestimate) and make the total num- ber of subatomic particles in the universe 10\80, or 10\IO\1.903.
Now, at least, we are starting with a double-ten number and that ought to do it. Skewes' number, here we come. All we have to do is take the superfactorial of ,10.80, something we can express as (l0\80)(10\80).
Working tliat out (and I hope I'm doing it cor- rectly), we get 10\10\81.9, or 10\10\10\1.91.
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And that lifts us into the "triple-ten numbers" for the first time. In fact, if we compare the superfac- torial of the total number of subatomic particles in the universe (which is IO'VIO'YIO^I.91) -with Skewes* number (which, as a triple-ten number, ' is IOVIO^JOV.34), we might think we were almost there.
We need to begin with something more than the number of subatomic particles in the universe—how about the amount of space in the universe?
The smallest unit of space we can conveniently deal with is the volume of a neutron, a tiny globe that is about 10'\ —13 centimeters in diameter, or one ten- trillionth of a centimeter.
The observable universe has a radius of 12.5 bil- lion light-years, or 1.25 X 10\10 light-years, and each light-year is equal to just under 10\l3 kilo- meters. Hence, the observable universe has a ra- dius of roughly 10\23 kilometers. Since 1 kilo- meter = 100,000, or 10\5, centimeters, the observable universe has a radius of roughly 10X28 centimeters. From this we can calculate the volume of the observ- able universe to be roughly equal to 4.2 X 10\84 cubic centimeters.
A neutron, with a diameter of ION.—13 centimeters, has a volume that is equal to roughly 5 X 10\ — 40 cu- bic centimeters. That means that the volume of the observable universe is -roughly 2XlO\124, or 10.124.3 times the volume of a single neutron.
Suppose we call the volume of space equal to that of a neutron a "vacuon." We can then say that there are 10X124.3 vacuous in the universe and call that the "vacuon-number."
The vacuon-number is nearly a billion billion bil-
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lion billion billion times greater than the number of subatomic particles in the universe, so we can feel pretty confident about the superfactorial of the vacuon-number, which is (10^124.3)^(10^124.3). except that this comes out to 10\10\1Q\2.10...
Despite the vastly greater quantity of empty space than of matter in the universe, -the rightmost compo- nent of the triple-ten number went up only from 1.91 to 2.10, with 34 as the goal. That's enough to depress us, but wait- In considering the number of vacuons in the uni- verse, we imagined it as existing at a moment in time. But time moves, and the universe changes. A sub- atomic particle that occupies one place at one moment may occupy another place at another moment. The most rapidly moving particles are, of course, the mass- less ones which move at the speed of light.
The speed of light is Just about 3 X 10'\10 centime- ters per second, and the smallest distance one can move with some significance is the diameter of a neu- tron, which is 10\ —13 centimeters. A photon will flash the width of a neutron, then, in about 3 X 10\ —24 seconds. We can consider this the small- est unit of time that has physical meaning and call it the "chronon."0
To imagine a long period of time, let's consider what we can call the "cosmic cycle," one period of ex- pansion and contraction of the universe (assuming it is oscillating). Some have guessed the length of the cosmic cycle to be 80,000,000,000, or 8 X 10\10, years.
- Stanley G. Weinbaum once imagined ;>pace and time quantized in this fashion in one of his science fiction stories and med the word "chronoii" for his ultimate particle of time.
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The number of chronons in one cosmic cycle, then, is roughly l0\42.
In every chronon of time, the universe is slightly different from what it was in the preceding chronon or what it will be in the next chronon, because,'if nothing else, every free-moving photon, neutrino, and graviton has shifted its position by the width of one neutron in some direction or other with each chronon that passes.
Therefore we might consider the total number of vacuons not only in the present universe, but also in the one that existed in the last chronon, the one that will exist in the next chronon, and, in general, all the universes in all the chronons through a cosmic cycle, (To be sure, the expansion and contraction of the uni- verse alters its vacuon content—these increasing in number with expansion and decreasing with contrac- tion—but we can suppose that the present size of the universe is about average.)
In that case, then, the total number of vacuons through every chronon of the cosmic cycle is just about l<r\ 166.3. What this means is that if you wish to place a proton somewhere in the universe at some instant in time, you have (under the conditions I've described-) a choice of 10'\166.3 different positions.
But if you take the superfactorial of this enor- mous "total-vacuon number," you end up with 10\10\10\2.27.
We have hardly moved. I Just can't seem to move those triple-ten numbers and make progress toward Skewes* number. I am Skewered.
In fact, it's worse tlian that. According to Mr. Boas, Skewes' determination of Skewes' number depended on the supposition that something called the "Rie-
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mann- hypothesis" is true. It probably is, but no one has proved it to be so.
In 1955 Skewes published a paper in which he cal- culated the value of the number below which the number of primes must be higher at some point than the formula would predict, if the Riemann hypothesis were not true.
tt turns out _that the Riemann-hypothesis-not-true case yields a number that is far higher than Skewes' number. The new number, Or what I suggest we call the Super-Skewes number, is 10\10\10\1000, or 10\10\10\10\3.
The Super-Skewes number and Skewes' number are both quadruple-ten numbers-10\10\10\10\3 and 10\10\10\10\1.53 respectively-and the differ- ence in the rightmost component seems to be small. However, vou saw what difficulty there was in budg- ing the triple-ten numbers upward. Well, moving the quadruple-ten numbers upward is far harder still, and Skewes' number is virtually zero in comparison to the Super-Skewes number.
If I had reached Skewes' number, I would still have had the Super-Skewes number ahead of me. I would have been Super-Skewered.
PART 4
PHYSICS
My major work, as far as physics is concerned, is my three-volume Understanding Phvsics, which is in- cluded among my first hundred books. Once that was done, there was little I could do in physics but forage about the edges of the subject and approach a differ- ent audience.
Among the small books for eight-year-olds that I wrote for Follett, and mentioned earlier, there is one book on physics— Light (Boo^ 208). From that book, here is the description of the spectrum: