It was partly due to my study of these figures, which was inspired by Buckminster Fuller, that I discovered highly composite numbers on my own.

Primordial Numbers

Buckminster Fuller introduces a class of numbers somewhat related to highly composite numbers which he called Scheherazade numbers. Although he never formally defines these, we can glean the fact that they are equal to the product of primes. (He called these Scheherazade numbers as the prime numbers 7 x 11 x 13 equal 1001 and Scheherazade was a character in One Thousand and One Nights.) Mathematicians call these numbers "prime-factorial" or "primordial" numbers. As examples, 1 x 2 x 3 x 5 gives the primordial number 30, and 1 x 2 x 3 x 5 x 7 x 11 x 13 gives the primordial number 30030.

However, the lower highly composite number 24 has as many factors as 30. And the primordial number 30030 has 64 factors, while the closest highly composite number, 27720, a lower number, has 96 factors, or 32 more factors. Although primordial numbers are more encompassing with regard to primes, highly composite numbers are more encompassing with regard to all numbers.

Applications of Highly Composite Numbers

Although Ramanujan brought these numbers to the attention of the mathematical community, has these numbers found their way into the mainstream of society? Have the public and legislators used these numbers to bring about ease of computation, packing and distribution? The answer is "no," (which is why I am writing this paper).

Hardy, (see Collected Papers of G. W. Hardy, Vol. VII), called Ramanujan's paper "... the largest and perhaps the most important connected piece of work which he has done since his arrival in England." Although Ramanujan died in 1920, at a young age, Hardy went on to live many more years, until 1947, and continued to study Ramanujan’s work. I wondered why Hardy did not try to apply this work to society. Hardy himself answered this question: "If asked to explain how, and why, the solution of the problems which occupy the best energies of my life is of importance to the general life of the community, I must decline the unequal contest ... A pure mathematician must leave to happier colleagues the great task of alleviating the sufferings of humanity." (quoted in The Man Who Knew Infinity, p. 347)

Highly composite numbers have been studied rather extensively by mathematicians. But perhaps they have been buried in with mountains of other mathematical facts, which may or may not have any practical value.

I think it is time that economists, psychologists, sociologists, legislators, and others become aware of these numbers.

Grouping by Dozens

The ancient merchants possibly intuitively knew to use a highly composite 12, or dozen, in their trade. There are many other examples of the ancients using a highly composite dozen or a close relative.

A base twelve numbering system was proposed in 1586 by Simon Stevin, and again in 1760 by Georges Louis Leclerc.

Then, in October of 1934, an author by the name of F. Emerson Andrews, wrote an article for The Atlantic Monthly, “An Excursion in Numbers,” which eventually led to the formation, in 1944, of the Duodecimal Society (later called the Dozenal Society). The Society rightly points out that a dozenal base is better than a ten base. It has investigated the mathematics of a dozenal system. It also publishes papers concerned with dozenal systems. I was the annual guest speaker for the Dozenal Society in 1997 and gave a lecture on “Numbers of the Future?,” a base twelve number system.

Isaac Asimov was a member of the Dozenal Society and has several paragraphs in Realm of Numbers concerning the advantages of a base twelve number system, but seems convinced that our ten fingers are too big an obstacle to overcome. This point might be refutable by simply counting with the thumb on the three bony segments of each four fingers, which gives a highly composite twelve.

Despite a few dissenters, the world has continued to slip toward base ten and the so-called scientific base ten metric system. For example, in 1971, the British switched from half-pennies, pennies, threepence, sixpence, shillings, half-crowns, pounds, and guineas (a 1/2, 1, 3, 6, 12, 30, 240, 252, system which uses numbers with lots of factors or highly composite numbers), to a decimal (1, 5, 10, 20, 50, 100, 1000, etc.) mostly non-highly composite, monetary system. The more highly composite numbers were in use when London was the foremost financial center of the world. The English system of measurement with 12 inches to the foot was in use when the U.S. put a man on the moon. Now the U.S. is trying (with much resistance) to go to a non-highly composite metric system.

It's a noble goal to align all your measuring systems with your number system, and those who have tried to do so should be thanked for their efforts. However, instead of changing our highly composite measuring system to match our numbers, perhaps we should have changed our numbers to match our more highly composite measuring system.

Perhaps we really just got off on the wrong foot (maybe I should say hand) when we started counting by tens, using our eight fingers and our two thumbs.

Base Ten Glasses

Why haven't highly composite numbers, in a sense the "shelters," the "great cathedrals," the “marvelous organisms” of all our numbers, been more intensively taught to the general public? I believe we've been completely surrounded by highly composite numbers for so long (360 degrees in a circle, 360 degrees in the heavens, 12 months, 60 seconds, 60 minutes, 24 hours, 12 in a dozen, 12 inches in a foot, 6-packs, 12-packs, 12-24-36 pictures in a roll of film, 12 notes, 12 pence, etc.) that we have forgotten about them. It’s like the story of two fish having an argument about whether water exists.

And even if someone does discover highly composite numbers, they are an embarrassment to a society that uses "ten," not a highly composite number, as the core of its number system. One can group numbers by |||||||||| (10, as we do), by |||||||||||| (12, as I suggest), or by |||||||||||||||||||||||| (20, as the Mayans did), or by |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| (60, as the Babylonians did), or whatever you want.

Various Ways to Mentally Group Numbers -- The Base of Various Number Systems

Highly composite numbers have more possible patterns than any smaller number. One interesting possibility would be to have a counting system that kept changing to the next highly composite number base. Our time system, which shifts from a 60 second and 60 minute base to a 24 hour base is somewhat like this. I think it might be possible.

The mental number pattern that Homo sapiens imposes on the world results in everyone looking at the world through a pair of glasses that could be called, “base-ten glasses.” These glasses will tend to filter out certain things, and let in other things. Someone with these glasses would see |||||||||||||||||||||||| and say, “I see ten and ten and four, or 24.” Someone with more highly composite base-twelve-filter-glasses would say, “I see twelve and twelve, or two dozen.”

THE FUNDAMENTAL PATTERNS OF A NUMBER

end Part II of III

Highly Composite Numbers
Part III
The Largest Highly Composite Numbers

Building a car that breaks the world’s land speed record can give insight into automotive design, and by the same logic, finding the largest highly composite can give insight into numbers.

I am not a computer programmer by trade, but I knew enough to program my old computer to calculate up to the one hundred and fifth highly composite, V(105). This was just beyond what Ramanujan had calculated without a computer. He went to V(103). Others have used computers to calculate highly composites far beyond this.

An abbreviated notation is used for writing these large numbers where “...” means “all the primes left out.” For example, highly composite number 1646, or V(1646) = 2^11 * 3^8 * 5^4 * 7^3 * (11...23)^2 * 29 *...241. (Using graphing calculator notation here, * means “times” and ^ means “to the power of.”)

A French paper was brought to my attention, “Methods d’Optimisation pour un Probleme de Theorie des Nombres,” by Guy Robin, in which he gives the first highly composite number to have more than 10^1000 (that’s a 1 followed by 1000 zeros) factors. This number is: 2^20 * 3^12 * 5^8 * 7^7 * (11*13*17)^5 * (19*23)^4 * (29...71)^3 * (73...421)^2 * 431...30113. It has 2^3203*3^68*5^2*7*13 factors. They reported that if we were to write it out it would have 13,198 digits. (However, Guy Robin infomed me recently by e-mail that it actually has 13,199 digits.)

Two mathematicians I met on the Internet, Noam Shazer and Matt Conroy, brought to my attention the method of generating highly composites of huge sizes by using primes. (See Appendix A.) For example, to calculate V(103) 6,746,328,388,800 one has to only know the primes up to 23. Highly composites of billions of digits become feasible given simply enough computing time and computing speed. Although this method can “sling shot” highly composites from primes, it does not find every highly composite.

Jud McCranie holds some records for generating a certain class of prime numbers. Using this method, he started programming his computer to generate highly composite numbers, and was able to produce a highly composite of 7 trillion digits

There is not a very concise way to write a large highly composite, that I know of, as there is with very large primes. For example a Mersenne prime with 909,562 digits can be written quite concisely as: 2^3,021,377 - 1. (This was the largest known prime, the last time I checked, but may not be any more. A Mersenne prime is just a prime of the form “2 to some power minus 1.” So, 2 to the 3rd power, minus 1, is 7, a prime.)

A computer programmer from Canada, John Boyer, sent me a proof of the infinitude of highly composite numbers. (Appendix D) Since Euclid proved that primes are infinite, John uses this to show that highly composites are infinite also.

The Hidden Chaotic Structure in Numbers

Primes are essentially inevitable but unpredictable. In other words, you always know another one is coming, but you never know exactly where it will appear.

Imagine a Neanderthal who somehow figured out that the first primes were 2, 3, 5, 7, 11, and 13. Now he is hunting for the next one. There is no way he can predict when it will arrive unless he tries. Is it 14? No, that’s divisible by 2. Is it 15? No, that’s divisible by 3. Is it 16? No, that’s divisible by 2. Is it 17? That’s not divisible by 2, by 3, by 4, or by any other number. It is 17.

I quote here from Tom M. Apostle, professor emeritus at Cal Tech, “no simple formula exists for producing all the primes.” In my thinking, a simple formula involves a pattern, or repetition. Any repetition implies multiplication. Multiplication implies a factor. And, if there is a factor involved, then, of course, we know that the number is not prime. There is no recipe.

This same rule, inevitable but unpredictable, holds true also for highly composite numbers.

They are inevitable, in that they are infinite, and they are unpredictable, in that we never know when the next one is coming. Just like with primes you have to try and see.

This rule is built into the fundamental nature of mathematics, in the similar way that the speed of light appears to be an upper boundary to the speed of moving objects. You can group by any base you want, and it doesn’t affect the hidden chaotic structure in numbers as given by primes and highly composites.

If you examine number theory in general, you find that it does not have much of a structure. John Stillwell, Mathematics and its History (p. 27): “number theory has never been submitted to a systematic treatment like that undergone by elementary geometry in Euclid’s Elements.”

So there is this sort of hidden chaos in numbers, with the primes on one side and the highly composites on the other side.

Actually, our misplaced faith in the complete regularity of numbers is somewhat a result of the base we use to group these numbers.

We can kind of get a feel for this chaos if we look at the following chart. In the first three columns, we see the numbers, primes, and highly composites in our Hindu-Arabic base-ten system. In the second three columns, we see them in their purity, without the filter-glasses-tool of a base-ten-grouping system.

Synergy, Wholism, and Systems Theory

With regard to highly composite numbers, there is a loose parallel to the history of psychology. Sigmund Freud and others intently studied psychotic and neurotic behavior, or aberration. This, of course, gave a skewed view of human nature. Abraham Maslow decided to study healthy people or what he called "self-actualized" people. This resulted in a new branch of psychology called humanistic psychology. The two together provided us with a more balanced view of humanity. In other words, I believe that the general public has been taught about numbers with a minimum of factors (primes), while they have not been taught about numbers with a large number of factors (highly composites).

Highly composite numbers may have potential use in the field of (w)holism. Wholism can be defined as “the theory that whole entities, as fundamental components of reality, have an existence other than as the mere sum of their parts.”

Wholism has been attacked by science philosopher Karl Popper and others. But Systems Analysis, a practical application of wholism, has been used successfully for many years. It was found that environmental problems did not respond to compartmentalized attacks, but needed a larger viewpoint. Harry J. White, in the Encyclopedia Americana, say Systems Analysis is “an outgrowth of advanced technology,” because of the “demands of society for more effective solutions of complex environmental, production, and transportation problems.”

Also, as Bucky Fuller used to point out, when one considers the alloy chrome-nickel-steel, it’s strength is not merely the sum of its individual parts. The primary components are iron, chromium, and nickel, with minor components of carbon, manganese, and others. The tensile strength of iron is about 60,000 pounds per square inch. Chromium is about 70,000 p.s.i.. Nickel is about 80,000 p.s.i. Carbon and the other minor components add another 50,000 p.s.i. Add them all up, and you get about 260,000 p.s.i. But the tensile strength of chrome-nickel-steel runs to about 350,000 p.s.i.

The number 12, due to its highly versatile nature, due to the structure of numbers, does appear to have an existence other than as the mere sum of its parts. (12 = 11+1, 10+2, 9+3, 8+4, 7+5, 6+6.) The number has more factors than numbers up to twice its sum.

Pure reductionists may say that we can still analyze why 12 is a unique number, (because it has more factors than any smaller number) but we must remember that  highly composite numbers, like primes, are unpredictable. If pure reductionism were completely valid, then we would expect that we could predict the next highly composite number.

For that matter, if pure reductionism were valid, then why couldn’t we predict the next prime number? And since we can’t even predict the next prime number, in the apparently abstract and well-ordered discipline of mathematics, can we expect a purely reductionistic approach to work in the real world of science?

I am sort of glad that we have this unpredictability in numbers. It seems to satisfy some inherent desire that the universe not be completely predictable by science.

Philosophically, I have long since known that a completely predictable universe would be a completely boring universe. We would have no interest in it.

Douglas Hofstadter, in his notable work, Gödel, Escher, Bach, also discusses “chaos in the midst of the most perfect, harmonious, and beautiful of all creations: the system of natural numbers.” (p. 398-408)

Of course there is order in numbers. 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ... It is obvious and simple. Yet this simple order, so apparent in counting and addition, somehow creates a sort of chaos in multiplication and division.

At some points, where the highly composites occur, there seems to be a sort of self-organization. One can not help but wonder if this mirrors the self-organization that occurs in living systems.

Perhaps we can generalize to: all things can be broken apart into their basics, yet all things are a part of infinitely greater structures.

Is the universe less a machine and more a living system, or organism (Newtonian paradigm to Gaian paradigm)?  Besides the fundamentals in nature, should we also emphasize the connections in nature?

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is probably the foundation of what structure there is in Number Theory. This states that every number can be broken down into a different set of prime numbers.

However, opposed to this, but connected to it, we might also say that every number is unique part of an infinite set of highly composite numbers. (Proof in Appendix E) Perhaps this should be called the Organic Theorem of Arithmetic. On the one hand, each number can be broken down into its parts, on the other hand each number is part of some greater structure that is nested within still greater structures, ad infinitum. (This theorem has also been proven.)

Every number is nested within highly composite numbers.

Highly composite, or versatile, numbers are numbers in which as many different parts as possible are interwoven into a whole, a unity. Unity + versatility = Uni-versal numbers.

If you think about number from this perspective an interesting thing happens. The figure becomes the ground, and the ground becomes the figure.

Highly Composite Life Systems

Let’s imagine again the human body (called by some a complex adaptive system) as an loose analogy. It can be broken down chemically into its elements (primes): carbon, oxygen, hydrogen, etc. These elements (primes) can be combined in many different ways that are completely unremarkable (non-highly composite numbers), but when they are combined into certain patterns, a human body (highly composite number) results. No one could have predicted it. But there it is. Perhaps it was inevitable.

The heart is a part of larger and larger systems. It is nested within them. The circulatory system, the body, the family, the community, the nation, all living things, the whole earth, the solar system, etc. Sort of like the highly composite number 8 is a part of the  highly composite numbers 24, 48, 120, 240, 360, 720, 360, 840, 1680 and every remaining highly composite number.

But the heart is not a part of the skeletal system, which is a larger system than the heart. Sort of like the number 8 is not part of every larger highly composite number. It is not a part of 12 or 36 for example.

Is it a mere coincidence that carbon, the basis of all life as we know it, is element number 6, with 6 protons, 6 neutrons, and 6 electrons? Perhaps. Is it a coincidence that Carbon 60, the spherical molecule of great strength, has a highly composite 60 number of atoms? Perhaps. Is it a coincidence that the primary fuel of the human body, C6H12O6, uses highly composite numbers? Perhaps. Is it a mere coincidence that DNA, the molecule that carries the blueprint of our bodies uses a highly composite 4 different parts? Perhaps.

Summary

An important class of numbers, equal in importance to prime numbers, possibly known to the ancient Babylonians, probably known about by Plato, was rediscovered by Ramanujan 1915.

These numbers have more potential built-in patterns than any smaller number. Numbers with no inherent patterns we call prime numbers (except for 1).

Mathematicians, in the past, as well as in recent times, have recommended using one of these highly composite numbers, 12, for our number base. However, thus far, society has stuck with the more awkward base ten system.

In 1992, I independently developed a base 12 number system and symbolism. In 1995, I independently rediscovered Ramanujan's “highly composite numbers” and dubbed them "versatile numbers."

Things can be shared evenly easier with these numbers than with other numbers. Thus, they have unexplored ramifications in economics, business, psychology, and other social sciences.

A specific name, "versatile numbers," was coined, to allow the majority of people to understand the importance of these numbers, and to raise the status of these numbers to that of at least the status of “prime” numbers.

POSSIBLE NUMBER NOMENCLATURES

Currently almost 6 billion people think, subconsciously, in terms of 10s and 100s, etc., not a very versatile system. Force a human mind to think in terms of 10s and 100s, when the most highly composite of all numbers are 2, 6, 12, 60, 360, and 2520, and you have perhaps to some degree limited that mind, and possibly even warped that mind.

I sometimes try to imagine what would be the synergistic effect if humankind’s collective numerical mind-pattern were made aware of highly composite numbers, and were eventually brought into synchronization with a highly composite number system. For example, what would happen if we had more highly composite, “base-twelve filter glasses?”

Probably people would continue to fight with each other, no matter what. However, numbers are a tool. Why not use the best possible tool to lubricate Homo sapiens economic and social interaction?

Conclusions

Homo sapiens routinely uses a mental-tool-filter-glasses-pattern to deal with continually increasing amounts in the world around him or her.
2. The mental-tool-filter-glasses-pattern predominantly used on Earth is probably |||||||||| or ten.
3. Many amounts (highly composite amounts) have relatively high potential patterns built-in.
4. Homo sapiens might benefit from becoming more aware of these more highly composite amounts. (2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720,...)
5. Homo sapiens might benefit from using a mental-tool-filter-glasses pattern with relatively high potential built-in patterns, as his or her primarily mental-tool-filter-glasses-pattern for dealing with the world around him or her.

Fundamental Thesis: Liberal use of highly composite numbers (2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, ...) could lubricate a human’s interaction with surrounding humans and the surrounding universe, especially in an increasingly populated world.

Additional minor theses:
a) The unpredictability of prime numbers and highly composite numbers suggests a mathematical foundation for chaos in nature.
b) The unpredictability of prime numbers and highly composite numbers may oppose the idea of a completely reductionist approach to science.
c) Highly Composite Numbers suggests a mathematical foundation for self-organization.

APPENDIX A: Sling Shot Method for determining some Highly Composite Numbers using Prime Numbers

This method was used by Ramanujan, and was brought to my attention by Noam Shazeer and Matt Conroy. (Note: Internet notation is used. * means “times”, ^ means “to the power of,” and _ means “subscript.”)

For any positive integer n, let f(n) denote the number of factors of n. If n = (p_1^e_1)(p_2^e_2)... (p_m^e_m) where p_i denotes the ith prime number, then the factors of n will be numbers of the form (p_1^f_1)(p_2^f_2)...(p_m^f_m) where 0<=f_i<=e_i for all i.

Since there are e_i +1 choices for each f_i, the number of factors of n is (e_1+1)(e_2+1) ... (e_m +1)

To generate a highly composite number, pick an arbitrary positive real number r. Now let us find the value(s) of n which maximize (f(n)^r)/n. If we find such an n, we can be sure that there is no x=f(n). If there were, (f(x)^r)/x would surely be greater than (f(n)^r)/n. So n would be a highly composite number.

Now let us find an n which maximizes (f(n)^r)/n.
Begin by noting that f(n)^r)/n = ((e_1+1)^r)/(p_1^e_1)*((e_2+1)^r)/(p_2^e_2) ... ((e_m+1)^r)/(p_m^e_m).

I will omit the proof that (f(n)^r)/n achieves a maximum. This will become obvious in a minute.

To achieve our desired n, we must, for each i, pick an e_i which maximizes ((e_i+1)^r/(p_i^e_i) > (((e_i-1)+1^r)/(p_i^(e_i-1)) if and only if
((e_i+1)/(e-i))^r > p_i if and only if
((e_i+1)/(e_i)) > (p_i)^(1/r) if and only if
1+1/(e_i) > (p_i)^(1/r) if and only if
(e_i) > 1/((p_i)^(1/r))-1)

So ((e_i+1)^r)/(p_i^e_i) is clearly maximized when e_i is the greatest integer less than 1/((p_i)^(1/r)-1). This will be positive only when p_i < 2^r, i.e. only finitely often.
So let us try our method,

Let r=4,
p_1e2, so e_1 = | 1 / ((2) ^ (1/4)-1)) | = 5
p_2=3, so e_2 = | 1 / ((3) ^ (1/4)-1)) | = 3
p_3=5, so e_3 = | 1 / ((5) ^ (1/4)-1)) | = 2
when p_i = 7, 11, and 13, e_i = 1.
when p_1 > 16, e_i = 0
So, n=2^5 * 3^3 * 5^2 * 7 * 11 * 13 = 21621600
f(n) = 576

Let r=6
p_1=2, so e_1 = | 1 / ((2) ^ (1/6)-1)) | = 8
p_2=3, so e_2 = | 1 / ((3) ^ (1/6)-1)) | = 4
p_3=5, so e_3 = | 1 / ((5) ^ (1/6)-1)) | = 3
p_4=7, so e_4 = | 1 / ((7) ^ (1/6)-1)) | = 2
p_5=11, so e_5 = 1 / ((11) ^ (1/6)-1)) | = 2
when p_i = 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, e_i = 1
when p_i > 64, e_i = 0
So n=2^8 * 3^4 8 5^3 * 7^2 * 11^ 2 *
13*17*19*23*29*31*37*41*43*47*53*59*61
which has 30 digits.
f(n) = 13271040*

To produce a highly composite number of about d digits, let r be approximately ln(d*ln(10))/ln(2). You should be able to compute the highly composite number in about d*ln(d) time.

Note: this method does not produce all highly composite numbers, e.g. no value of r produces the highly composite number 4.

APPENDIX B: Teaching Highly Composite Numbers

First, one would want the students to know how to factor a number. There are certain helpful rules that are found in many mathematics textbooks, or which, I believe could be derived by the students with proper guidance. Some examples of these rules are: 1) all even numbers can be divided by two, 2) all numbers that end in 5 or 0 can be divided by five, 3) all numbers whose digits add to a number divisible by 3 can be divided by 3. There are other rules, and one could go into as much depth as one wished.

Second, have the students use this knowledge of factoring to make a factor table as shown at the beginning of this chapter. They may need some assistance with some of the primes, but they should be able to make a table up to say 60 or 120 (depending on their grade level).

Third, have them circle all the primes. That is, have them circle all the numbers with just two factors. One could then also go into as much detail concerning primes as one wished. Last, have them copy the table over (or use the same table), and have them circle, in the last column, every time a number is greater than all the previous numbers. In other words, (looking at the first column now), every time a number had more factors than all the previous numbers. Explain that these numbers are highly composite numbers and give an example of how they are more highly composite than other numbers, especially primes. The example of 23 students in a class versus 24 students in a class might be a good example to use. Of course, a more advanced class could use a computer program to generate highly composite numbers.

There may be better ways to teach this.

APPENDIX C: Possible Time-Line To Convert to a Highly Composite Number System

This section has been removed.

APPENDIX D: Proof of the Infinitude of Highly Composite Numbers

Consider a highly composite number V1, and compute the next prime number P that is greater than V1. Now consider the number V2 = PV1. The number V2 is obviously greater than V1. More importantly, V2 has at least one more factor than V1, namely P. The proof of this is simple: P cannot be a factor of V1 since V1 < P, and all other factors of V1 must be factors of V2 because any factor F such that FG = V1 would appear as F(GP) = V2.

Thus, since V2 has more factors than V1, the next highly composite number after V1 must be either V2 or appear before V2 numerically. This establishes the existence of a highly composite number that is greater than any given highly composite number. Inductively, this implies that the set of highly composite numbers is unlimited provided we have an infinitude of primes from which to draw P, which was established by Euclid.

Submitted by John Boyer (jboyer@uwi.com)

APPENDIX E: Proof that every number, N, is a factor of some highly composite number, V

Thm1: (proved by Ramanujan, 1915): Let a[p][k] denote the exponent of the prime p in the k-th V number; For any fixed p, when k goes to infinity, a[p][k] tends to infinity.

Thm 2: Let V be a highly composite number with prime factorization
V = PROD (i = 1 to inf, p_i ^ a_i).
Clearly, a_i = 0 for sufficient i.
Now, suppose a_i < a_(i+1). Then by replacing p_i^a_i and p_(i+1)^a_(i+1)
with p_i^a_(i+1) and p_(i+1)^a_i, respectively, in the product for V, we
obtain a V' < V with the same number of factors. This contradicts the fact that V was highly composite. We conclude that a_i >= a_(i+1), which is to say, the exponents in the prime factorization of V occur in descending order. This means that the 0 exponents occur after the positive exponents. Thm 3: Every positive number N is a factor of some highly composite number. Proof: Every finite number N has a prime factorization (fundamental theorem of arithmetic). Consider the highest exponent E from the primes in N's factorization, and consider the highest prime P appearing in N's factorization. By Thm 1, there exists a highly composite V[k] for which P^E is afactor (the power of P may be greater than E in V[k]). By Thm 2, everyprime less than P appears in the factorization of V[k] with exponents of Eor higher. Hence, N is a factor of V[k].

By J. L. Nicolas, John Boyer, and Dave Wilson.

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