Primes Peter Martinson To begin the investigation of Gauss s early work, we must begin with prime numbers. People are usually taught that numbers come out of counting how many objects there are. We all know how to count you just add 1 each time, right? Count up to twenty. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Pythagoras and his school defined several different classes of these numbers, according to their properties. For example, only half of our set of numbers can be divided in half evenly, the other half will have 1 left over. Thus, 4 can be divided into 2 + 2, and 14 into 7 + 7, but 11 can only be divided in half if it is first reduced by one, 11 1 = 5 + 5. The Pythagoreans called the evenly divisible numbers even (for obvious reasons), and the other ones odd. Some of the even numbers, when divided into two equal parts, divide into two even numbers (like 4), some into two odd numbers (like 14). These are called evenly-even and unevenlyeven, respectively. Each of the odd numbers can be similarly classified, when you subtract 1 from them. So, 11 would be an unevenly-odd number, and 13 an evenly-odd. Let s rearrange our numbers a bit: Wait, those aren t numbers! Even if numbers are thought of as collections of objects, those collections can be broken up and rearranged into neat geometric shapes. There are actually three types of structure represented here: the square, the rectangle, and the long, skinny rectangle. The long, skinny ones are called the prime numbers, because they can t be broken down neatly in any way. Try to find some way of predicting the character of the next number, without building it with the blocks, and then see if you were right. For example, how far are the square numbers from each other? Is there a pattern? Now, try to predict when the next prime number will occur. This isn t so easy. After getting up to about 41, the determination of whether a number is prime or not becomes more difficult. This is not just because most people don t own more than 41 blocks 1. The ancient Greek scientist Eratosthenes developed a method to single out the primes, which is called his sieve. Attach a black marker to a circle, then roll that circle on a piece of paper in a straight line. Each time it rotates once, it leaves a mark on the paper. Now, do the same with a circle of twice the diameter, but using a different colored marker. This will leave a mark 1 You could go out and buy sugar cubes, or even raisins! 1
for every two of the first dots. Now, do it with a circle three times the diameter. Eratosthenes claimed that, each black dot that has more than one other color on it will represent a composite number (like 8). If it only has one other color on it, then it represents a prime number (like 3). Each number that is not prime, is divisible into several equal columns, which may also be divided further, until only prime numbers are left. For example, 6 can be divided into two groups of three - both prime numbers. 8 can be divided into two groups of four, or two sets of two groups of two. We can write this as 6 = 2 3, and 8 = 2 2 2. Or, take 24. It can be divided into either four groups of six, or three groups of eight, or two groups of 12, like this: Each of those groups can be divided also, until we have reduced 24 to 2 2 2 3. Notice that each composite number is composed of only smaller prime numbers. For example, 41 is not composed of any prime factors (because it is prime itself!), but there is no number smaller than 41 that has 41 as a factor. Now that we have located our prime number, though, a whole new class of number is possible those that have 41 as a factor, such as 41 2 = 82, 41 3 = 123 or 41 187 = 7667. Thus, if we call our generic prime number p, then no number less than p will indicate that p is prime. On the other hand, the whole field of numbers after p is shaped by p s existence. This can be thought of in a different way, which is called the method of inversion, and is how Gauss thought. The existence of the numbers above p make this number necessary. The distribution of the primes is thus determined from above, not below. It is determined by the infinite, not from the building blocks. Let us state this in a different way: How many prime numbers are there? All numbers, as we ve seen, are either prime numbers, or composed of prime numbers. If there were assumed to be a last prime number, then the number of primes would be finite, and every number after that would be composed of primes. If we multiply all of the primes together, then we make one huge, composite number: A. There is no reason why we can t add 1 to this huge, composite number, to get A + 1 = B. Since we re already way past the last prime number, this new number, made out of the composite number plus 1, must also be a composite number. But, here s the paradox we ve already used all the prime numbers to make A, the huge composite number. Therefore, A + 1 = B has to contain some other prime number as a factor, one that s not on our finite list of prime numbers. Therefore, there will always be at least one more prime number than any finite set of prime numbers, and thus, there is no last prime number. But, no number before any prime number p will give any hint at p s existence. Therefore, you can t calculate the next prime number either. This leaves us with a paradox: If none of the numbers before a prime number give any hints about the existence of that prime number, then those primes could just as well be scattered 2
about randomly! This was a paradox that confounded empiricists like Euler, who was tormented by trying to calculate the next prime number. Gauss thought differently about the universe of number. Logarithmic Interlude When Gauss was 15, he was introduced to the Duke of Brunswick by his future teacher, E. A. W. von Zimmermann. The Duke was so impressed by this young man, that he agreed to be his patron, and pay for his attendance at thc Collegium Carolinum. Soon after this meeting, the Duke s minister of state delivered to young Gauss a new copy of Schultze s Table of Logarithms, which included a list of the prime numbers up to 10,009. Gauss immediately did what any normal, creative kid would do, began extending the tables of logarithms, and made a discovery. The logarithm is a concept introduced by John Napier of Scotland, but whose roots lie in Plato s Greece. The later Pythagoreans, including Plato and Archytas, emphasized the study of mean proportionals, which they called the science of Arithmetic. They noted several types of mean proportional, such as the arithmetic mean and the geometric mean. The arithmetic mean is the half-way point between two magnitudes. You can create an arithmetic progression by adding a fixed amount over and over, like we do when we count normally. For example, the series 1, 5, 9, 13, 17, 21, 25, 29, etc., is an arithmetic series beginning at 1, and increasing by 4 each time. Each of the terms in an arithmetic series can be represented by the expression an + b, where b is the initial value, a is the value of the increment, and n is some integer. Thus, all terms in the previous progression are represented by 4n + 1. If a is 7 and b is 4, all numbers in the progression are represented by 7n + 4, which are 4, 11, 18, 25, 32, 39, 46, etc. The geometric mean is related to the smaller extreme in the same way as the larger extreme is related to the mean. This is famously demonstrated by the doubling of the square, which, when done repeatedly, generates a simple geometric progression. Thus, we get areas 1, 2, 4, 8, 16, 32, and so forth. Each term in the geometric progression, is represented by the power of some base. In the doubling progression, we start with a base of 2, and each term is equal to 2 n, where n is the number of times the doubling has occurred. Hence, 2 0 = 1 (because the square hasn t been doubled yet), 2 1 = 2, 2 2 = 4, etc. A closer look at the physical construction of the doubling squares, will reveal that it actually involves both types of growth. It is the changes in both area and side length that follow a geometric progression, but each rotation is equal. Therefore, the rotation grows arithmetically. Like the prime numbers before, this construction grows in steps. What we are looking for here, is some continuous function that determines this growth, as if from above. 3
Of course! It s nothing other than Bernoulli s Logarithmic spiral. Here, Bernoulli combines the two types of growth into one. As the angle grows at a constant rate, the distance to the curve from the center grows at a geometric rate. Notice that both types of growth occur at right angles to each other. The logarithm of a number is merely the amount of rotation needed to get to a certain place on the spiral. Leibniz also studied this perpendicular, orthogonal, relationship. Instead of the arithmetic growth being circular and the geometric being radial, however, Leibniz had the arithmetic grow horizontally, while having the geometric grow vertically. Today, the curve produced is called Exponential, and its inverse is called Logarithmic. If you really attempted to construct these curve, you ran into a problem. It is easy to line up the geometrically related heights at equal intervals, but connecting the tops of them with a smooth curve is just guesswork. The geometric and arithmetic progressions are discrete, while the curves they supposedly describe are continuous. Seemingly, one could get a closer approximation to the curve by using smaller and smaller intervals, but there would always be some space between the tips of the lines where the height is not exactly known. So, what is the relationship between the curve and the discrete scaffolding? Let us apply Gauss s method of inversion. Suppose, instead of constructing the geometric 4
and arithmetic progressions, the curve is constructed first. 2 By now marking out the arithmetic progressions along the horizontal line, the geometrically growing heights can be found immediately. And, any arithmetic interval can be chosen! Both of those progressions are thus the discrete expressions of the higher, continuously changing logarithmic function. In other words, look at the two forms of progression, geometric and arithmetic, as being caused by the principle of logarithmic growth. The two forms of progression were created by the logarithm, in the same way that the prime numbers were made necessary by some principle that is infinite to the field of counting numbers. We have the objects of sense perception (the progressions, and the primes), but we also have some unsensed, higher principle ordering the objects of sense perception, which is reaching in and organizing things from outside the domain of the senses. Gauss s first discovery As Gauss learned how logarithms worked, he also investigated the nature of the prime numbers at the back of his new book. Even though he was 15 years old, he knew that the Creator of the Universe had not strewn his creation about randomly, but that He had organized it harmonically. Gauss began counting the prime numbers. He looked at how fast the primes accumulate, as your set of numbers is expanded. Then, he looked at how fast the number of those numbers grew, which have only two factors. Then, only three factors. He soon found that 10,009 contained too few prime numbers for his experiments, so he calculated more. As he said later on in life: I have very often employed a spare unoccupied quarter of an hour in order to count up a chiliad here and there; however, I eventually dropped it completely, without having quite completed the first million. 3 Gauss was not interested in the next prime number, per se, but was interested in how they were distributed. How does the composition sound, as a whole? On the back page of his book of logarithms, he wrote: Gauss 1791 Prime numbers under a (= ) Numbers with two factors with 3 factors, probably and so forth to infinity. 1 a L a L L a a, L a 2 (L L a)2 a L a, (where L stands for the natural logarithm) What is this? Let s analyze these formulas, to see what they mean. First, log a means, given some radius a on your spiral, how much rotation did it take to get there? The first formula is a proportion, where the number of prime numbers under a is to 1, as a (the radius of the spiral) 2 Sky Shields has invented a machine that can construct both the exponential curve, and Bernoulli s spiral, continuously. See his report in the January 2008 issue of υναµις. 3 Gauss s letter to Johann Franck Encke, December 24, 1849 5
is to its logarithm. As the spiral angle increases at a constant rate, the growth of the radius accelerates, since it is growing geometrically. Conversely, if the radius is made to increase at a constant rate, the change in the angle must appear to slow down. The initial slow-down of the angular growth will be rapid, but the rate of change of the slow-down will also slow down. Therefore, the ratio will appear to start steep, but then level off to a somewhat linear growth. What did Gauss see? Under 10, there are four prime numbers: 2, 3, 5, 7. But, if you double the range, the number of primes also doubles, adding 11, 13, 17, and 19. If we continue this, how many primes should be beneath 100? 40, right? Wrong, there are only 25, because the frequency slows down. For example, there are only two primes from 20 to 30. The rate of accumulation must therefore slow down. We can make a diagram of this, where we count out the numbers along a horizontal line, and place a mark above each number which represents the number of primes beneath that number. You will find that chart below, compared with the rate of growth of Gauss natural logarithm function: They look almost identical! Here, they are superimposed: So, they re not exactly equal. As will be seen with more study of Gauss s work, he never relied on a formula, though his papers are absolutely full of them. He performed experiments, developed hypotheses, tested them with more experiments, and constantly changed his hypotheses until his death. 4 What does this mean? Is that higher principle which orders the prime numbers, the same principle which generates the arithmetic and geometric progressions? Five years before his death, Gauss wrote a letter to his student Johann Franck Encke about his hypotheses on prime numbers. The young astronomer had just sent Gauss his own hypotheses of the distribution of the primes. Gauss had recognized that, no matter how the primes were grouped, their distribution would always be inversely proportional to the natural logarithm. 4 Unfortunately, he only published his hypotheses in public after he felt he had a bulletproof argument for their validity. 6
Here is a graph showing how many primes are between each successive group of a thousand numbers. The red curve is 1 Log n. In this letter, Gauss also discussed the work of other scientists, such as Lagrange, who had come up with a similar distribution. He gave a suggestion for improving the distribution formula, but said that judging the accuracy of the hypothesized distribution required yet more experimental data, and also required proofreading of the tables of primes then in use, such as Lambert s error-riddled table. Gauss called for the extension of the prime number tables to several million, and suggested hiring a young calculating prodigy, Johann Martin Zacharias Dase (1824-1861) 5, for the job. Less than a decade later, a young student of Gauss, Bernhard Riemann, gave his first lecture at Göttingen University. It was a lecture on the distribution of prime numbers, and he asserted that the distribution hypothesized by Gauss would always get closer and closer to the actual distribution of the prime numbers. This has been known since then as the Riemann Hypothesis, and scientists have been fighting to verify it ever since. Some still argue that, since Riemann hasn t demonstrated the hypothesis for the last prime number, then it can t be asserted as true! 5 The ability to perform calculations is not a measure of genius, although it is one of those special abilities that can be used for the good. From what I have been able to dig up on Dase, it appears that he sold himself as a type of circus act, performing huge calculations for the entertainment of others. Gauss saw that this ability could be used for the advancement of human knowledge, instead of for entertainment, and thus established Dase s immortality. Dase did end up constructing a book of all the factors for numbers between 7,000,000 to 10,000,000, but died before he could do much more. 7