## Quadratic Reciprocity## Peter Martinson## Gauss lied, almostDisquisitiones Arithmeticae - actually, he provides two. As he points out, this was the first time that any proof at all had been stated publicly. The nature of mathe-matical proof is somewhat mystified today, not in small thanks to how Gauss set forth his proofs. The way Gauss discovered those principles that he published in all his public works, is not the same as how he explained them. He was quite concerned with 1) not revealing his true, Keplerian method of scientific work, and 2) ensuring that what he did discover was rendered bulletproof against all possible slings and arrows thrown from a British Fascist dominated scientific community.^{1} It was politically dangerous to announce radical breakthroughs in science, whose discovery necessitated methods that differed from the pure empiricism of Euler, Lagrange, and Laplace. Therefore, Gauss's only defense, since he could not keep all of his discoveries secret without going nuts, was to present them as results of mathematically logical proofs.
proof that the discovery made is actually a true, valid principle that is efficient in the Creator's universe. The method of discovery, itself, is proof, which includes reference to an experiment whose success depends on the existence of that principle of the universe. What Gauss presented to his scientific audience, was thus a smokescreen, which, more often than not, did not include either the crucial experiment, or the description of how Gauss came to discover that, for which he was presenting proof. For example, Gauss recognized the validity of the principle of Quadratic Reciprocity before he completed his first proof in 1796. His proof does not resemble how he discovered that it was valid. This is also true for his next 7 proofs of the same theorem! He says as much in his introduction to his third proof:
The questions of higher arithmetic often present a remarkable characteristic which seldom appears in more general analysis, and increases the beauty of the former subject. While analytic investigations lead to the discovery of new truths only after the fundamental principles of the subject (which to a certain degree open the way to these truths) have been completely mastered; on the contrary in arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries... ## Leonhard "Turncoat" Euler^{2} Euler thence participated in taking Leibniz's beautiful ideas, one by one, and degrading their meaning.
Disquisitiones Arithmeticae were due first to Euler, as also to Fermat, Lambert, Lagrange, and Legendre. But, Gauss says in the introduction of that work, that
... [A]s one result led to another I had completed most of what is presented in the first four sections of this work before I came into contact with similar works of other geometers. ... disprove discoveries, but it does not prove the validity of any discovery. Gauss's method was, first, to make the discovery, and then to write down a bunch of mathematical derivations that would appear to corroborate the discovery. Euler was just the opposite, and spent his time doing mathematical derivations, hoping that the mind of the creator would be revealed to him through that arduous task. He went blind in at least one eye while doing these types of calculations.
The fundamental theorem must certainly be regarded as one of the most elegant of its type. No one has thus far presented it in as simple a form as we have done above. This is even more surprising since Euler already knew other propositions which depend on it and from which it can easily be recovered... [A]ll his attempts at demonstration were in vain, and he succeeded only in giving a greater degree of verisimilitude to the truth that he had discovered by induction.Gauss then gives several examples of Euler's work in this direction, and then the futile attempts of Euler's follower, Napoleon Bonaparte's "Great Volcano of the Mathematical Sciences," Joseph Louis Lagrange. ## Pythagorean Arithmetic^{2}+b^{2}=c^{2}, where c is the length of the hypotenuse of a triangle whose other sides, a and b, meet at a right angle.
sine and cosine, continuously proceed through every possible length between 0 and 5 cm. Now, draw four other triangles with hypotenuse 5 cm, with one leg being successively 1 cm, 2 cm, 3 cm, 4 cm. We will call that leg a.
^{2}=1 is equal to 5^{2}=25. In other words, 25-1=24 is the square of the length of that side. Now, we break out our trusty old calculator and calculate √ 24, which is approximately 4.8989795... (of course, Pythagoras couldn't do it like this, but that's OK).
always be one of these infinite decimals? Let's try one more, a=3. Ah! Now, we get 25-9=16, which is a square number. This triangle is what we will call the 3-4-5 triangle.
continuum of possibilities. We have selected one instant out of a continuous change, represented by the circle animation above. In this sense, the length 5 has a special property. The reader will find that, not all whole number hypotenuses are capable of being in a right triangle with two legs of whole number lengths.
Pythagorean Triples, because of the emphasis Pythagoras, himself, placed on them. How many sets of Pythagorean Triples are there? There is the 3-4-5 triangle, and all of its multiples, such as the 6-8-10 triangle. But, these triangles are all similar to each other. Another Pythagorean Triple is the triangle with legs 8, 15, and hypotenuse 17.
transcendental difference between the circle and the straight line.^{5}
the inverse problem.^{6} A number of singularities, selected out of some continuous process, can be accessed by human experimental work. Only the human mind is capable of conjecturing what that continuous domain must be, which would generate those singularities. If a person discovers the nature of that continuous domain - that universal physical principle - he can apply that knowledge to generate any singularity he wishes. He has thus gained a power.
gnomon, or an "L" shape with the 1 at the intersection, as in the animation below. Now, the gnomon is surrounding another square shape, whose side length is 24. We've thus just constructed two new squares, of side lengths 24 and 25. Hence, we have three lengths that would construct a right triangle, 7^{2}+24^{2}=25^{2}. The crafty reader should figure out how to begin with an even square, instead of an odd.
## Interlude on FermatDisquisitiones Arithmeticae. Usually, Gauss cites Fermat as the originator of the most beautiful theorems in the book, and then usually says that either Euler tried to prove them, or did prove them. One of these most important theorems relates to our development of the Pythagorean Triples, and the curious relationship between square numbers and the primes.
^{2}+4^{2}=5^{2}. Most of the time, when adding two square numbers, we get a number that is not square, such as 3^{2}+5^{2}=9+25=34, which is 2 less than the nearest square. 34 itself is a composite number, being made up of 17·2. Let us develop the sum of every pair of square numbers up to 100:
two that are true square numbers, 25 and 100. The others are either composite numbers, or prime numbers. Let us collect the prime numbers:
all of the 4n+1 numbers below 200.^{7} In other words, those prime numbers that would be part of a Pythagorean Triple, were they square numbers, are those primes who have -1 as a quadratic residue! Fermat claimed to have proved that all of the prime numbers of the form 4n+1 are the sum of two squares, and that all the rest of the primes are not. Hence, we have another geometric insight into these primes:
all 4n+1 primes perform the same function as the square of the hypotenuse of a Pythagorean Triangle, and also cause -1 to function as a special type of square number.Perhaps they are not prime numbers at all, but themselves a type of square number, for which the concept of number, itself must be expanded.
This will have profound implications later, when we look at Gauss's treatment of Biquadratic Residues, but let this suffice as a lead-in to the principle of Quadratic Reciprocity.
## Reciprocityinversion: It is relatively simple to calculate all the quadratic residues of a given modulus, but, to find all those moduli for which a given number is a quadratic residue, is much more difficult. What is the principle which determines the distribution of quadratic residues among the moduli?
which are themselves quadratic residues of +3. For example, 7 is a quadratic residue of 3, and -3 is congruent to 4 (mod 7), which is obviously a quadratic residue. Our circle of modulus 3 will show that, only those numbers congruent to 1 modulus 3 will be quadratic residues, which are all equal to 3n+1 for some n (not including 3 itself). In other words, all prime numbers of the form 3n+1 have -3 as a quadratic residue, and if they are also of the form 4n+1, then +3 is also a quadratic residue. On the other hand, if the 3n+1 number is also a 4n+3 number, then +3 is not a quadratic residue. This is complete.
inductive investigation, and begins constructing a general principle. Let a and a¢ be two numbers of the form 4n + 1, and let b and b¢ be numbers of the form 4n + 3. If a is a quadratic residue of a¢, then we will write aRa¢, and if a is a quadratic nonresidue of a¢, we will write aNa¢. We saw that when aNa¢ (or, what is the same, -aNa¢), then a¢Na and -a¢Na. When +aRb, then -aNb (by anti-symmetry), but both +bRa and -bRa. Conversely, when both +bNa and -bNa, then +aNb and -aRb. When +bRb¢ (-bNb¢), then +b¢Rb (-b¢Nb). If you observe the relationships between more moduli, you will see that what we have found seems to always be the case.
Fundamental Theorem, "[s]ince almost everything that can be said about quadratic residues depends on this theorem."^{8} It is today known as the Principle of Quadratic Reciprocity. With a bit of deduction, all of the reciprocal relationships can now be determined. Here is Gauss's chart of relationships:
^{2}=361.
residue of the 4n+3 quadratic nonresidues. The reader should try these cases.
^{2}. 19 should then be a quadratic nonresidue of all of its 4n+3 quadratic residue. So, for modulus 7, 19 is congruent to 5, which is indeed a quadratic nonresidue of 7.
residue. This is true for 31, where 19 is congruent to 9^{2}=81.
Golden Theorem. Principles of the universe are rarely simple to state, yet they are always efficient.
## The Proofs of Gausswhy did Gauss value his golden theorem so much, that he needed to prove it eight times? And, why did the others, such as Euler, not consider it that important? Because Gauss was searching for principles governing the organization of the universe, experimentally!
A fifth method of proving theThe very next entry (mid-September 1801) states: A new, quite simple and very convenient method for investigating the elements of the orbital motions of celestial bodies.He immediately applied this "very convenient method" to the determination of the orbit of Ceres. Disquisitiones Arithmeticae. Take one of our modulus circles, such as that for 17, and draw a diameter through the 0 point. Rotate a radius up to the first quadratic residue, 1, and draw in its sine. Do this for all the quadratic residues. Your circle should look like this:
residue, but will become quadratic residues if 2 is a quadratic nonresidue.
whenever k is a quadratic nonresidue of the modulus, otherwise, it stays positive.
elegant relationship between the transcendental functions of the circle and the quadratic residues in order to give his fifth proof of the validity of the principle of Quadratic Reciprocity. Is this how he discovered it? No. But it does give some insight into how his mind worked.
The State of Our Union: The End of Our Delusion!, Lyndon LaRouche referred to Gauss's emphasis on the principle of Quadratic Reciprocity:
All consistent mathematics as such, reflects obviously underlying ontological, axiomatic-like presumptions, which, however strenuously "pure" mathematicians may attempt to hide this fact, are "secretions" rooted in the physical geometry inherent in the Complex Domain. This investigation can wait until the pedagogicals on Gauss's work on Biquadratic Residues.
## Footnotes:^{1}See Lyndon LaRouche's paper, "[cite the paper]"
^{2}See David Shavin's article, etc.
^{3}Disquisitiones Arithmeticae § 151
^{5}It is for this reason, that Lyndon LaRouche once poked at his friends that, since 5 is one of these special numbers, it is actually an irrational number!
^{6}Gauss himself called it "Indefinite Analysis."
^{7}notice, also, that 25 is a 4n+1 number
^{8}Disquisitiones Arithmeticae, p. 88
^{9}The State of our Union: The End of our Delusion!, Lyndon H. LaRouche, Jr., p. 81, LaRouche PAC, 2007 File translated from T _{E}X
by T_{T}H,
version 3.80.On 04 Mar 2008, 22:59. |