Part 7 (2/2)

When you magnify a spot, What you had before, you've got.

Smaller, smaller, smaller, yet, Still the same details are set; Finer than the finest hair Blake's infinity is there, Rich in structure all the way- Just as the mystic poets say.

Some of the modern applications of the Golden Ratio, Fibonacci numbers, and fractals reach into areas that are much more down to earth than the inflationary model of the universe. In fact, some say that the applications can reach even all the way into our pockets.

A GOLDEN TOUR OF WALL STREET.

One of the best-known attempts to use the Fibonacci sequence and the Golden Ratio in the a.n.a.lysis of stock prices is a.s.sociated with the name of Ralph Nelson Elliott (18711948). An accountant by profession, Elliott held various executive positions with railroad companies, primarily in Central America. A serious alimentary tract illness that left him bedridden forced him into retirement in 1929. To occupy his mind, Elliott started to a.n.a.lyze in great detail the rallies and plunges of the Dow Jones Industrial Average. During his lifetime, Elliott witnessed the roaring bull market of the 1920s followed by the Great Depression. His detailed a.n.a.lyses led him to conclude that market fluctuations were not random. In particular, he noted: ”the stock market is a creation of man and therefore reflects human idiosyncrasy.” Elliott's main observation was that, ultimately, stock market patterns reflect cycles of human optimism and pessimism.

On February 19, 1935, Elliott mailed a treatise ent.i.tled The Wave Principle The Wave Principle to a stock market publication in Detroit. In it he claimed to have identified characteristics which ”furnish a principle that determines the trend and gives clear warning of reversal.” The treatise eventually developed into a book with the same t.i.tle, which was published in 1938. to a stock market publication in Detroit. In it he claimed to have identified characteristics which ”furnish a principle that determines the trend and gives clear warning of reversal.” The treatise eventually developed into a book with the same t.i.tle, which was published in 1938.

Figure 125

Figure 126 Elliott's basic idea was relatively simple. He claimed that market variations can be characterized by a fundamental pattern consisting of five waves during an upward (”optimistic”) trend (marked by numbers in Figure 125 Figure 125) and three waves during a downward (”pessimistic”) trend (marked by letters in Figure 125 Figure 125). Note that 5, 3, 8 (the total number of waves) are all Fibonacci numbers. Elliott further a.s.serted that an examination of the fluctuation on shorter and shorter time scales reveals that the same pattern repeats itself Figure 126 Figure 126), with all the numbers of the const.i.tuent wavelets corresponding to higher Fibonacci numbers. Identifying 144 as ”the highest number of practical value,” the breakdown of a complete market cycle, according to Elliott, might look as follows. A generally upward trend consisting of five major waves, twenty-one intermediate waves, and eighty-nine minor waves (Figure 126) is followed by a generally downward phase with three major, thirteen intermediate, and fifty-five minor waves (Figure 126).

Figure 127 Some recent books that attempt to apply Elliott's general ideas to actual trading strategies go even further. They use the Golden Ratio to calculate the extreme points of maximum and minimum that can be expected (although not necessarily reached) in market prices at the end of upward or downward trends (Figure 127). Even more sophisticated algorithms include a logarithmic spiral plotted on top of the daily market fluctuations, in an attempt to represent a relations.h.i.+p between price and time. All of these forecasting efforts a.s.sume that the Fibonacci sequence and the Golden Ratio somehow provide the keys to the operation of ma.s.s psychology. However, this ”wave” approach does suffer from some shortcomings. The Elliott ”wave” usually is subjected to various (sometimes arbitrary) stretchings, squeezings, and other alterations by hand to make it ”forecast” the real-world market. Investors know, however, that even with the application of all the bells and whistles of modern portfolio theory, which is supposed to maximize the returns for a decided-on level of risk, fortunes can be made or lost in a heartbeat.

You may have noticed that Elliott's wave interpretation has as one of its ingredients the concept that each part of the curve is a reduced-scale version of the whole, a concept central to fractal geometry. Indeed, in 1997, Benoit Mandelbrot published a book ent.i.tled Fractals and Scaling in Finance: Discontinuity, Concentration, Risk Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, which introduced well-defined fractal models into market economics. Mandelbrot built on the known fact that fluctuations in the stock market look the same when charts are enlarged or reduced to fit the same price and time scales. If you look at such a chart from a distance that does not allow you to read the scales, you will not be able to tell if it represents daily, weekly, or hourly variations. The main innovation in Mandelbrot's theory, as compared to standard portfolio theory, is in its ability to reproduce tumultuous trading as well as placid markets. Portfolio theory, on the other hand, is able to characterize only relatively tranquil activity. Mandelbrot never claimed that his theory could predict a price drop or rise on a specific day but rather that the model could be used to estimate probabilities of potential outcomes. After Mandelbrot published a simplified description of his model in Scientific American Scientific American in February 1999, a myriad of responses from readers ensued. Robert Ihnot of Chicago probably expressed the bewilderment of many when he wrote: ”If we know that a stock will go from $10 to $15 in a given amount of time, it doesn't matter how we interpose the fractals, or whether the graph looks authentic or not. The important thing is that we could buy at $10 and sell at $15. Everyone should now be rich, so why are they not?” in February 1999, a myriad of responses from readers ensued. Robert Ihnot of Chicago probably expressed the bewilderment of many when he wrote: ”If we know that a stock will go from $10 to $15 in a given amount of time, it doesn't matter how we interpose the fractals, or whether the graph looks authentic or not. The important thing is that we could buy at $10 and sell at $15. Everyone should now be rich, so why are they not?”

Elliott's original wave principle represented a bold if somewhat naive attempt to identify a pattern in what appears otherwise to be a rather random process. More recently, however, Fibonacci numbers and randomness have had an even more intriguing encounter.

RABBITS AND COIN TOSSES.

The defining property of the Fibonacci sequence-that each new number is the sum of the previous two numbers-was obtained from an unrealistic description of the breeding of rabbits. Nothing in this definition hinted that this imaginary rabbit sequence would find its way into so many natural and cultural phenomena. There was even less, however, to suggest that experimentation with the basic properties of the sequence themselves could provide a gateway to understanding the mathematics of disordered systems. Yet this was precisely what happened in 1999. Computer scientist Divakar Viswanath, then a postdoctoral fellow at the Mathematical Sciences Research Inst.i.tute in Berkeley, California, was bold enough to ask a ”what if?” question that led unexpectedly to the discovery of a new special number: 1.13198824 .... The beauty of Viswanath's discovery lies primarily in the simplicity of its central idea. Viswanath merely asked himself: Suppose you start with the two numbers 1, 1, as in the original Fibonacci sequence, but now instead of adding the two numbers to get the third, you flip a coin to decide whether to add them or to subtract the last number from the previous one. You can decide, for example, that ”heads” means to add (giving 2 as the third number) and ”tails” means to subtract (giving 0 as the third number). You can continue with the same procedure, each time flipping a coin to decide whether to add or subtract the last number to get a new one. For example, the series of tosses HTTHHTHTTH will produce the sequence 1, 1, 2, 1, 3, 2, 5, 3, 2, 5, 7, 2. On the other hand, the (rather unlikely) series of tosses HHHHHHHHHHHH... will produce the original Fibonacci sequence.

In the Fibonacci sequence, terms increase rapidly, like a power of the Golden Ratio. Recall that we can calculate the seventeenth number in the sequence, for example, by raising the Golden Ratio to the seventeenth power, dividing by the square root of 5, and rounding off the result to the nearest whole number (which gives 1597). Since Viswanath's sequences were generated by a totally random series of coin tosses, however, it was not at all obvious that a smooth growth pattern would be obtained, even if we ignore the minus signs and take only the absolute value of the numbers. To his own surprise, however, Viswanath found that if he ignored the minus signs, the values of the numbers in his random sequences still increased in a clearly defined and predictable rate. Specifically, with essentially 100 percent probability, the one hundredth number in any of the sequences generated in this way was always close to the one hundredth power of the peculiar number 1.13198824..., and the higher the term was in the sequence, the closer it came to the corresponding power of 1.13198824 .... To actually calculate this strange number, Viswanath had to use fractals and to rely on a powerful mathematical theorem that was formulated in the early 1960s by mathematicians Hillel Furstenberg of the Hebrew University in Jerusalem and Harry Kesten of Cornell University. These two mathematicians proved that for an entire cla.s.s of randomly generated sequences, the absolute value of a number high in the sequence gets closer and closer to the appropriate power of some fixed number. However, Furstenberg and Kesten did not know how to calculate this fixed number; Viswanath discovered how to do just that.

The importance of Viswanath's work lies not only in the discovery of a new mathematical constant, a significant feat in itself, but also in the fact that it ill.u.s.trates beautifully how what appears to be an entirely random process can lead to a fully deterministic result. Problems of this type are encountered in a variety of natural phenomena and electronic devices. For example, stars like our own Sun produce their energy in nuclear ”furnaces” at their centers. However, for us actually to see the stars s.h.i.+ning, bundles of radiation, known as photons, have to make their way from the stellar depths to the surface. Photons do not simply fly through the star at the speed of light. Rather, they bounce around, being scattered and absorbed and reemitted by all the electrons and atoms of gas in their way, in a seemingly random fas.h.i.+on. Yet the net result is that after a random walk, which in the case of the Sun takes some 10 million years, the radiation escapes the star. The power emitted by the Sun's surface determined (and continues to determine) the temperature on Earth's surface and allowed life to emerge. Viswanath's work and the research on random Fibonaccis that followed provide additional tools for the mathematical machinery that explains disordered systems.

There is another important lesson to be learned from Viswanath's discovery-even an eight-hundred-year-old, seemingly trivial mathematical problem can still surprise you.

I should attempt to treat human vice and folly geometrically...the pa.s.sions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature. ... I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes and solids.-BARUCH S SPINOZA (16321677) (16321677)Two and two the mathematician continues to make four, in spite of the whine of the amateur for three, or the cry of the critic for five.-JAMES M MCNEILL W WHISTLER (18341903) (18341903) Euclid defined the Golden Ratio because he was interested in using this simple proportion for the construction of the pentagon and the pentagram. Had this remained the Golden Ratio's only application, the present book would have never been written. The delight we derive from this concept today is based primarily on the element of surprise. surprise. The Golden Ratio turned out to be, on one hand, the simplest of the continued fractions (but also the ”most irrational” of all irrational numbers) and, on the other, the heart of an endless number of complex natural phenomena. Somehow the Golden Ratio always makes an unexpected appearance at the juxtaposition of the simple and the complex, at the intersection of Euclidean geometry and fractal geometry. The Golden Ratio turned out to be, on one hand, the simplest of the continued fractions (but also the ”most irrational” of all irrational numbers) and, on the other, the heart of an endless number of complex natural phenomena. Somehow the Golden Ratio always makes an unexpected appearance at the juxtaposition of the simple and the complex, at the intersection of Euclidean geometry and fractal geometry.

The sense of gratification provided by the Golden Ratio's surprising emergences probably comes as close as we could expect to the sensuous visual pleasure we obtain from a work of art. This fact raises the question of what type of aesthetic judgment can be applied to mathematics or, even more specifically, what did the famous British mathematician G.o.dfrey Harold Hardy (18771947) actually mean when he said: ”The mathematician's patterns, like the painter's or the poet's, must be beautiful.”

This is not an easy question. When I discussed the psychological experiments that tested the visual appeal of the Golden Rectangle, I deliberately avoided the term ”beautiful.” I will adopt the same strategy here, because of the ambiguity a.s.sociated with the definition of beauty. The extent to which beauty is in the eye of the beholder when referring to mathematics is exemplified magnificently by a story presented in the excellent 1981 book The Mathematical Experience The Mathematical Experience by Philip J. Davis and Reuben Hersh. by Philip J. Davis and Reuben Hersh.

In 1976, a delegation of distinguished mathematicians from the United States was invited to the People's Republic of China for a series of talks and informal meetings with Chinese mathematicians. The delegation subsequently issued a report ent.i.tled ”Pure and Applied Mathematics in the People's Republic of China.” By ”pure,” mathematicians usually refer to the type of mathematics that at least on the face of it has absolutely no direct relevance to the world outside the mind. At the same time, we should realize that Penrose tilings and random Fibonaccis, for example, provide two of the numerous examples of ”pure” mathematics turning into ”applied.” One of the dialogues in the delegation's report, between Princeton mathematician Joseph J. Kohn and one of his Chinese hosts, is particularly illuminating. The dialogue was on the topic of the ”beauty of mathematics,” and it took place at the Shanghai Hua-Tung University.

Since, as this dialogue starkly indicates, there is hardly any formal, accepted description of aesthetic judgment in mathematics and how it should be applied, I prefer to discuss only one particular element in mathematics that invariably gives pleasure to nonexperts and experts alike-the element of surprise.

MATHEMATICS SHOULD SURPRISE.

In a letter written on February 27, 1818, the English Romantic poet John Keats (17951821) wrote: ”Poetry should surprise by a fine excess and not by Singularity-it should strike the Reader as a wording of his own highest thoughts, and appear almost a Remembrance.” Unlike poetry, however, mathematics more often tends to delight when it exhibits an unantic.i.p.ated result rather than when conforming to the reader's own expectations. In addition, the pleasure derived from mathematics is related in many cases to the surprise felt upon perception of totally unexpected relations.h.i.+ps and unities. A mathematical relation known as Benford's law provides a wonderful case study for how all of these elements combine to produce a great sense of satisfaction.

Take a look, for example, in the World Almanac World Almanac, at the table of ”U.S. Farm Marketings by State” for 1999. There is a column for ”Crops” and one for ”Livestock and Products.” The numbers are given in U.S. dollars. You would have thought that the numbers from 1 to 9 should occur with the same frequency among the first digits of all the listed marketings. Specifically, the numbers starting with 1 should const.i.tute about one-ninth of all the listed numbers, as would numbers starting with 9. Yet, if you count them, you will find that the number 1 appears as the first digit in 32 percent of the numbers (instead of the expected 11 percent if all digits occurred equally often). The number 2 also appears more frequently than its fair share-appearing in 19 percent of the numbers. The number 9, on the other hand, appears only in 5 percent of the numbers-less than expected. You may think that finding this result in one table is surprising, but hardly shocking, until you examine a few more pages in the Almanac Almanac (the numbers above were taken from the 2001 edition). For example, if you look at the table of the death toll of ”Some Major Earthquakes,” you will find that the numbers starting with 1 const.i.tute about 38 percent of all the numbers, and those starting with 2 are 18 percent. If you choose a totally different table, such as the one for the population in Ma.s.sachusetts in places of 5,000 or more, the numbers start with 1 about 36 percent of the time and with 2 about 16.5 percent of the time. At the other end, in all of these tables the number 9 appears first only in about 5 percent of the numbers, far less than the expected 11 percent. How is it possible that tables describing such diverse and apparently random data all have the property that the number 1 appears as the first digit 30-some percent of the time and the number 2 around 18 percent of the time? The situation becomes even more puzzling when you examine still larger databases. For example, accounting professor Mark Nigrini of the c.o.x School of Business at Southern Methodist University, Dallas, examined the populations of 3,141 counties in the 1990 U.S. Census. He found that the number 1 appeared as the first digit in about 32 percent of the numbers, 2 appeared in about 17 percent, 3 in 14 percent, and 9 in less than 5 percent. a.n.a.lyst Eduardo Ley of Resources for the Future in Was.h.i.+ngton, D.C., found very similar numbers for the Dow Jones Industrial Average in the years 1990 to 1993. And if all of this is not dumfounding enough, here is another amazing fact. If you examine the list of, say, the first two thousand Fibonacci numbers, you will find that the number 1 appears as the first digit 30 percent of the time, the number 2 appears 17.65 percent, 3 appears 12.5 percent, and the values continue to decrease, with 9 appearing 4.6 percent of the time as first digit. In fact, Fibonacci numbers are more likely to start with 1, with the other numbers decreasing in popularity (the numbers above were taken from the 2001 edition). For example, if you look at the table of the death toll of ”Some Major Earthquakes,” you will find that the numbers starting with 1 const.i.tute about 38 percent of all the numbers, and those starting with 2 are 18 percent. If you choose a totally different table, such as the one for the population in Ma.s.sachusetts in places of 5,000 or more, the numbers start with 1 about 36 percent of the time and with 2 about 16.5 percent of the time. At the other end, in all of these tables the number 9 appears first only in about 5 percent of the numbers, far less than the expected 11 percent. How is it possible that tables describing such diverse and apparently random data all have the property that the number 1 appears as the first digit 30-some percent of the time and the number 2 around 18 percent of the time? The situation becomes even more puzzling when you examine still larger databases. For example, accounting professor Mark Nigrini of the c.o.x School of Business at Southern Methodist University, Dallas, examined the populations of 3,141 counties in the 1990 U.S. Census. He found that the number 1 appeared as the first digit in about 32 percent of the numbers, 2 appeared in about 17 percent, 3 in 14 percent, and 9 in less than 5 percent. a.n.a.lyst Eduardo Ley of Resources for the Future in Was.h.i.+ngton, D.C., found very similar numbers for the Dow Jones Industrial Average in the years 1990 to 1993. And if all of this is not dumfounding enough, here is another amazing fact. If you examine the list of, say, the first two thousand Fibonacci numbers, you will find that the number 1 appears as the first digit 30 percent of the time, the number 2 appears 17.65 percent, 3 appears 12.5 percent, and the values continue to decrease, with 9 appearing 4.6 percent of the time as first digit. In fact, Fibonacci numbers are more likely to start with 1, with the other numbers decreasing in popularity in precisely the same manner as the just-described random selections of numbers! in precisely the same manner as the just-described random selections of numbers!

Astronomer and mathematician Simon Newcomb (18351909) first discovered this ”first-digit phenomenon” in 1881. He noticed that books of logarithms in the library, which were used for calculations, were considerably dirtier at the beginning (where numbers starting with 1 and 2 were printed) and progressively cleaner throughout. While this might be expected with bad novels abandoned by bored readers, in the case of mathematical tables they simply indicated a more frequent appearance of numbers starting with 1 and 2. Newcomb, however, went much further than merely noting this fact; he came up with an actual formula formula that was supposed to give the probability that a random number begins with a particular digit. That formula (presented in Appendix 9) gives for 1 a probability of 30 percent; for 2, about 17.6 percent; for 3, about 12.5 percent; for 4, about 9.7 percent; for 5, about 8 percent; for 6, about 6.7 percent; for 7, about 5.8 percent; for 8, about 5 percent; and for 9, about 4.6 percent. Newcomb's 1881 article in the that was supposed to give the probability that a random number begins with a particular digit. That formula (presented in Appendix 9) gives for 1 a probability of 30 percent; for 2, about 17.6 percent; for 3, about 12.5 percent; for 4, about 9.7 percent; for 5, about 8 percent; for 6, about 6.7 percent; for 7, about 5.8 percent; for 8, about 5 percent; and for 9, about 4.6 percent. Newcomb's 1881 article in the American Journal of Mathematics American Journal of Mathematics and the ”law” he discovered went entirely unnoticed, until fifty-seven years later, when physicist Frank Benford of General Electric rediscovered the law (apparently independently) and tested it with extensive data on river basin areas, baseball statistics, and even numbers appearing in and the ”law” he discovered went entirely unnoticed, until fifty-seven years later, when physicist Frank Benford of General Electric rediscovered the law (apparently independently) and tested it with extensive data on river basin areas, baseball statistics, and even numbers appearing in Reader's Digest Reader's Digest articles. All the data fit the postulated formula amazingly well, and hence this formula is now known as Benford's law. articles. All the data fit the postulated formula amazingly well, and hence this formula is now known as Benford's law.

Not all lists of numbers obey Benford's law. Numbers in telephone books, for example, tend to begin with the same few digits in any given region. Even tables of square roots of numbers do not obey the law. On the other hand, chances are that if you collect all the numbers appearing on the front pages of several of your local newspapers for a week, you will obtain a pretty good fit. But why should it be this way? What do the populations of towns in Ma.s.sachusetts have to do with death tolls from earthquakes around the globe or with numbers appearing in the Reader's Digest? Reader's Digest? Why do the Fibonacci numbers also obey the same law? Why do the Fibonacci numbers also obey the same law?

Attempts to put Benford's law on a firm mathematical basis have proven to be much more difficult than expected. One of the key obstacles has been precisely the fact that not all lists of numbers obey the law (even the preceding examples from the Almanac Almanac do not obey the law precisely). In his do not obey the law precisely). In his Scientific American Scientific American article describing the law in 1969, University of Rochester mathematician Ralph A. Raimi concluded that ”the answer remains obscure.” article describing the law in 1969, University of Rochester mathematician Ralph A. Raimi concluded that ”the answer remains obscure.”

The explanation finally emerged in 19951996, in the work of Georgia Inst.i.tute of Technology mathematician Ted Hill. Hill became first interested in Benford's law while preparing a talk on surprises in probability in the early 1990s. When describing to me his experience, Hill said: ”I started working on this problem as a recreational experiment, but a few people warned me to be careful, because Benford's law can become addictive.” After a few years of work it finally dawned on him that rather than looking at numbers from one given source, the mixture mixture of data was the key. Hill formulated the law statistically, in a new form: ”If distributions are selected at random (in any unbiased way) and random samples are taken from each of these distributions, then the significant-digit frequencies of the of data was the key. Hill formulated the law statistically, in a new form: ”If distributions are selected at random (in any unbiased way) and random samples are taken from each of these distributions, then the significant-digit frequencies of the combined sample combined sample will converge to Benford's distribution, even if some of the individual distributions selected do not follow the law.” In other words, suppose you a.s.semble random collections of numbers from a hodgepodge of distributions, such as a table of square roots, a table of the death toll in notable aircraft disasters, the populations of counties, and a table of air distances between selected world cities. Some of these distributions do not obey Benford's law by themselves. What Hill proved, however, is that as you collect ever more of such numbers, the digits of these numbers will yield frequencies that conform ever closer to the law's predictions. Now, why do Fibonacci numbers also follow Benford's law? After all, they are fully determined by a recursive relation and are not random samples from random distributions. will converge to Benford's distribution, even if some of the individual distributions selected do not follow the law.” In other words, suppose you a.s.semble random collections of numbers from a hodgepodge of distributions, such as a table of square roots, a table of the death toll in notable aircraft disasters, the populations of counties, and a table of air distances between selected world cities. Some of these distributions do not obey Benford's law by themselves. What Hill proved, however, is that as you collect ever more of such numbers, the digits of these numbers will yield frequencies that conform ever closer to the law's predictions. Now, why do Fibonacci numbers also follow Benford's law? After all, they are fully determined by a recursive relation and are not random samples from random distributions.

Well, in this case it turns out that this conformity with Benford's law is not a unique property of the Fibonacci numbers. If you examine a large number of powers of 2 (21 = 2, 2 = 2, 22 = 4, 2 = 4, 23 = 8, etc.), you'll see that they also obey Benford's law. This should not be so surprising, given that the Fibonacci numbers themselves are obtained as powers of the Golden Ratio (recall that the = 8, etc.), you'll see that they also obey Benford's law. This should not be so surprising, given that the Fibonacci numbers themselves are obtained as powers of the Golden Ratio (recall that the n nth Fibonacci number is close to Fibonacci number is close to ). In fact, we can prove that sequences defined by a large cla.s.s of recursive relations follow Benford's law. ). In fact, we can prove that sequences defined by a large cla.s.s of recursive relations follow Benford's law.

Benford's law provides yet another fascinating example of pure mathematics transformed into applied. One interesting application is in the detection of fraud or fabrication of data in accounting and tax evasion. In a broad range of financial doc.u.ments, data conform very closely to Benford's law. Fabricated data, on the other hand, very rarely do. Hill demonstrates how such fraud detection works with another simple example, using probability theory. In the first day of cla.s.s in his course on probability, he asks students to do an experiment. If their mother's maiden name begins with A through L, they are to flip a coin 200 times and record the results. The rest of the cla.s.s is asked to fake a sequence of 200 heads and tails. Hill collects the results the following day, and within a short time he is able to separate the genuine from the fake with 95 percent accuracy. How does he do that? Any sequence of 200 genuine coin tosses contains a run of six consecutive heads or six consecutive tails with a very high probability. On the other hand, people trying to fake a sequence of coin tosses very rarely believe that they should record such a sequence.

A recent case in which Benford's law was used to uncover fraud involved an American leisure and travel company. The company's audit director discovered something that looked odd in claims made by the supervisor of the company's healthcare department. The first two digits of the healthcare payments showed a suspicious spike in numbers starting with 65 when checked for conformity to Benford's law. (A more detailed version of the law predicts also the frequency of the second and higher digits; see Appendix 9.) A careful audit revealed thirteen fraudulent checks for amounts between $6,500 and $6,599. The District Attorney's office in Brooklyn, New York, also used tests based on Ben-ford's law to detect accounting fraud in seven New York companies.

Benford's law contains precisely some of the ingredients of surprise that most mathematicians find attractive. It reflects a simple but astonis.h.i.+ng fact-that the distribution of first digits is extremely peculiar. In addition, that fact turned out to be difficult to explain. Numbers, with the Golden Ratio as an outstanding example, sometimes provide a more instantaneous gratification. For example, many professional and amateur mathematicians are fascinated by primes. Why are primes so important? Because the ”Fundamental Theorem of Arithmetic” states that every whole number larger than 1 can be expressed as a product of prime numbers. (Note that 1 is not considered a prime.) For example, 28 = 2 2 7; 66 = 2 3 11; and so on. Primes are so rooted in the human comprehension of mathematics that in his book Cosmos Cosmos, when Carl Sagan (19341996) had to describe what type of signal an intelligent civilization would transmit into s.p.a.ce he chose as an example the sequence of primes. Sagan wrote: ”It is extremely unlikely that any natural physical process could transmit radio messages containing prime numbers only. If we received such a message we would deduce a civilization out there that was at least fond of prime numbers.” The great Euclid proved more than two thousand years ago that infinitely many primes exist. (The elegant proof is presented in Appendix 10.) Yet most people will agree that some primes are more attractive than others. Some mathematicians, such as the French Francois Le Lionnais and the American Chris Caldwell, maintain lists of ”remarkable” or ”t.i.tanic” numbers. Here are just a few intriguing examples from the great treasury of primes: The number 1,234,567,891, which cycles through all the digits, is a prime.

The 230th largest prime, which has 6,400 digits, is composed of 6,399 9s and only one 8. largest prime, which has 6,400 digits, is composed of 6,399 9s and only one 8.

The number composed of 317 iterations of the digit 1 is a prime.

The 713th largest prime can be written as (10 largest prime can be written as (10 1951 1951)X(101975 + 1991991991991991991991991)+1, and it was discovered in-you guessed it-1991. + 1991991991991991991991991)+1, and it was discovered in-you guessed it-1991.

From the perspective of this book, the connection between primes and Fibonacci numbers is of special interest. With the exception of the number 3, every Fibonacci number that is a prime also has a prime subscript (its order in the sequence). For example, the Fibonacci number 233 is a prime, and it is the thirteenth (also a prime) number in the sequence. The converse, however, is not true: The fact that the subscript is a prime does not necessarily mean that the number is also a prime. For example, the nineteenth number (19 is a prime) is 4181, and 4181 is not a prime-it is equal to 113 37.

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