## Primes## Peter Martinsoncreative, not linear. It is necessary to see how Gauss thought at the time. We have some hints about this, and we also have the work of Lyndon LaRouche, whose discoveries in physical economy point towards what was physically important in Gauss.
## Gauss knew what counts
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?
^{1}.
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 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).
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.
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.
## Logarithmic InterludeTable 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.
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.
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.
continuous function that determines this growth, as if from above.
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."
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?"
^{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 discoveryI 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. how they were distributed. How does the composition sound, as a whole? On the back page of his book of logarithms, he wrote:
(where L stands for the natural logarithm) ^{4}
^{5}, for the job.
last prime number, then it can't be asserted as true!
## Footnotes:^{1}You could go out and buy sugar cubes, or even raisins!
^{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 &Delta&upsilon&nu&alpha&mu&iota&sigmaf.
^{3}Gauss's letter to Johann Franck Encke, December 24, 1849
^{4}Unfortunately, he only published his hypotheses in public after he felt he had a bulletproof argument for their validity.
^{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. |