## Arithmetic to GeometryBy Michael Kirsch Earlier, we did a thorough investigation of the tables of least residues for real numbers to determine the characteristic division of complex moduli whose norms are either 8n+1 or 8n+5. Now that we have just discovered these properties in the complex plane, primarily for -1 and i, let's return to the tables for induction with other numbers. We'll start with the number 2. (Right click on the Animation below and select "Zoom In" for better visibility). ## The Number 2^{2} will therefore be a (primitive root)(primitive root) = quadratic residue for 8n+5, and either (quadratic residue)(quadratic residue) or (biquadratic residue)(biquadratic residue), which are both biquadratic residues, for 8n+1. But why is 2 not a quadratic residue of 8n+5? Does this have something to do with the fact that p-1, which always has a period of 2, is a biquadratic residue in 8n+1, and a quadratic residue in 8n+5? Or does it have something to do with the fact that in 8n+5, ±2 is always a primitive root, while ±2i is always a quadratic residue or biquadratic residue, and also, that -1 is always a quadratic residue residue of 8n+1, but always a Q 8n+5. Also, 1 and its associates quadratic residue and 2 and its associates are all quadratic residues, only in special cases of 8n+1, like 17 and 97, are. It happens to be the case, that period 4 is a factor of [(p-1)/4]. However, these considerations don't tell us why 2 is a biquadratic residue of some 8n+1 numbers and a quadratic residue for others.
would be the reason for why 2 has the character of a biquadratic residue for some 8n+1 numbers, but is a quadratic residue for others? How can we distinguish the numbers from each other?
First Treatise on Biquadratic Residues he shows for the number 2, that if 8n+1 numbers are broken up into the form of a^{2} + b^{2}, there is a distinct difference, in the value of b for those which 2 is a quadratic residue of, versus those for which it is a biquadratic residue.
Gauss writes the following:
"....the number 2 must be included in class A for all moduli for which b is of the form 8n, and on the contrary, in class C for all moduli for which b is of the form 8n+4. However this theorem demands a much deeper investigation than the one which we found in the foregoing chapter, and its proof must be prefaced with many preliminary discussions, which are related to the sequence in which the numbers of the complex A,B,C,D follow one another." " II. Let p be a prime number of the form 8n+1. I say that +2 and –2 will be biquadratic residues or non-residues of p, as p is or is not of the form xx+64yy. For example, among the numbers 17, 41, 73, 89, 97, 113, 137. You will only find 73 = 9+64, 89 = 25+64, 113 = 49+64, and 254 º 2 (mod.73), 54 = 2 (mod. 89), 204 = 2 (mod. 113)." ## Reaping the Harvest of Induction^{1}
complex moduli whose norms are 8n+1 and 8n+5. Remember that for every table of real number residues, there is a separate table that could be made with complex numbers. We have saved ourselves the tedium of constructing such a table, because we can go backwards.
it is taken as a modulus. Furthermore, 5 is congruent to the associates of both of these conjugates. That means 5 is congruent to 8 possible numbers! In the table here for modulus 61 i.e., 5+6i, 1+2i, its conjugate, and the associates of each of them, are shown. Therefore, if one says 61 is a biquadratic residue residue of 5, which 5 are we talking about?! And therefore, for reciprocity, which 5 would be the one relative to modulus 61?
not prime.
As we began to investigate [Cubic and Biquadratic residues]from 1805 on, besides the elements that first presented themselves, some special theorems were yielded, which are of the utmost prominence due to their simplicity as well as the difficulty of their proofs, [new sentence? ] however we soon came to the recognition that the hitherto employed principles of arithmetic would in no way suffice for the establishment of a general theory, and that rather this necessarily demands the domain of higher arithmetic to be infinitely more enlarged, so to speak. "Thus, induction also yields to us a rich harvest of special theorems, which are transformed from the theorem for the number 2; however a common thread is lacking, a strong proof is lacking, since the method by which we have dealt with the number 2 in the first treatise does not permit of a broader application. There is indeed no lack of various methods, by means of which the proofs for special cases can be obtained, in particular those which pertain to the distribution of the quadratic residues between the complexes A and C; [however we will not be satisfied with this, because we are obliged to desire a general theory which encompasses every case.] Having already begun to consider this subject in the year 1805, we soon came to the conviction thatIs this actually how Gauss went about his initial investigation? There is no doubt that the way in which Gauss composes the two treatises illuminates the distinction between real and complex numbers with his shining example of the limitation of induction for real numbers; however, it is still unclear how Gauss himself came upon the relation of complex numbers and the study of biquadratic residues. He emphasizes 1805 as the beginning of his study of biquadratic residues. That would imply that his employment of complex numbers before 1805 had not yet become completely unified with the study of biquadratic residues. character that seemingly leads to the need to introduce the complex modulus as the instrument for the this task. Are not the two therefore, inseparably intertwined?
Harmony of the World, he does in fact demonstrate the harmonic ordering of the planets, and thus a principle of gravitation defined by a maintenance of the harmonic organization of the system as a whole. But how does he do this? With what instrument? Did he not define the measurement of the harmonic proportions as taken from the mind's knowability? Did he not define what the mind finds harmonic as a priori measurements of the soul? This demonstrates the human mind's higher intrinsic measurement which unites sight and sound, as the way in which the universe is measured. Here, one might say, that the principle discovered, reflects the instrumentation used to discover it, and the significance of the new instrument leading to the discovery is a realization of the potential of the instrument. Here the instrument that Gauss is bringing into being, with the seemingly simple investigation of the biquadratic character of numbers, is the complex modulus, but, in a larger sense, it is a change in the way the nature of the mind conceives number.
a priori, which, as we've begun to get an image of, will only become clear once the full significance of the Ö{-1} is fully introduced.
a priori, as the only means to uncover the principle projecting the paradox into the domain of real numbers.
^{2}
a priori concepts used to measure with, and what we are measuring is inherently physical. The numbers are a priori, and we use them to measure with, until what we are measuring demands a new number. The field of complex numbers opens up an entire realm of physical functions which can now be comprehended, as seen for example in Gauss' work on elliptical function, and his Copenhagen prize essay. Therefore, in the Quadrivium, arithmetic and geometry are not separate compartments of the mind, but are seen as a continuous process inherent to the mind's relationship with the principles organizing space.
## Footnotes:^{1}It can be noted in this table that for pairs of 8n+5 numbers, and pairs of 8n+1 numbers, reciprocity for non-residues does hold, i.e., if one number is a non-residue of another, likewise the other will be non-residue of it; however, for 8n+1 and 8n+5 taken together, it does not.
^{2}See Leibniz' New Essay on Human Understanding, Chapter 1
^{3}These subjects are discussed elsewhere on this site. |