Arithmetic to Geometry

By 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 is a biquadratic or quadratic residue of 8n+1, but a primitive root of 8n+5. This immediately makes clear why 4 is a biquadratic residue residue of 8n+1 and a quadratic residue of 8n+5, since 22 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.
It appears the characteristic of the number 2 is more complicated, as we are not directly dealing with the periods of length 2 and 4 as we were with -1 and i and -i.
In fact, what we just came upon for the number 2 ourselves, is the way in which Gauss begins his first inductive investigation of determining the biquadratic character of a number. The question of determining the biquadratic character of the number 2 immediately divides numbers of the form 4n+1 into 8n+1 and 8n+5. Subsequently, the first treatise on biquadratic residues is almost entirely on the determination of the biquadratic character of the number 2. Let's hear from Gauss:
". . .If we subject the moduli of the form 8n + 1 to an inductive investigation, we find that 2 belongs to A [biquadratic quadratic residues] for p = 73, 89, 113, 233, 257, 281, 337, 353, . . ., and on the contrary, that 2 belongs to C [quadratic residues] for p = 17, 41, 97, 137, 193, 241, 313, 401, 409, 433, 449, 457, . . . It appears that no simple criterion presents itself, at least at first glance, according to which the modulus of the former type could differ from that of the latter type."
What 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?
From our former investigations, the first thing we might suspect is that perhaps their factors of p-1 have a characteristic difference. This doesn't appear to yield anything telling. However, think of how we began our discussion about the characteristic of 4n+1 numbers. They can be made into the sums of two squares, and the sum of a square and twice a square. This is where we must look for our answer. What we have come across here is that determining the biquadratic character of prime numbers demands going beyond simply the periods of p-1, into the form of the number itself.
In the Gauss' First Treatise on Biquadratic Residues he shows for the number 2, that if 8n+1 numbers are broken up into the form of a2 + b2, 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:
"Hence, [we present] here the decompositions of the numbers p for which 2 belongs to the classes"
+ 16
+ 16
+ 16
+ 16
+ 16
+ 16
"....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."
In April 30th 1807 Gauss, in a letter to Sophie Germain, writes that, to convey to her the "curiosity" of the study of biquadratic residues, an example is necessary. The example he gives is precisely the question of the character of number 2, and its inductive proof is exactly the one he shows inductively in the first treatise almost 25 years later!
" 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)."
However, the proof of the inductive theorem of how the sum of two squares indicates the class of 2 is far more complicated than this inductive observation concerning the revealing nature of the sums of two squares. Having noted this, we will move on to the odd primes, and save the proof of this inductive observation for a later pedagogical.

Reaping the Harvest of Induction

Now that we've throughly treated -1, i, and 2, what about the primes under 100?
Picture of the same
For the 4n+1 primes 5 through 97, there appears to be no common characteristic, however, perhaps we may find something in the reciprocity relations. Before, in an earlier section, the principle of quadratic reciprocity was discussed. For 4n+1 numbers, if one number is a quadratic residue relative to another number taken as modulus, then that number taken as modulus will have the other number as a quadratic residue as well. For example, since 5 is a quadratic residue mod 29, 29 is a quadratic residue mod 5.
Can we extend the law of quadratic reciprocity to one of biquadratic reciprocity? Let us see. Looking at the table, something doesn't work out. 5 is a quadratic residue mod 61, but 61 is a biquadratic residue mod 5. 41 is a biquadratic residue mod 5, but 5 is a quadratic mod 41. 41 is a biquadratic residue mod 73, but 73 is a quadratic residue mod 41. In almost all cases, if a number is a biquadratic residue of a given modulus, then that number taken as a modulus will have the other number as a quadratic residue.
But, we cannot state this as a invariant relation, because also in this table, between 13 and 53, 13 and 61, and 73 and 89, biquadratic reciprocity holds, i.e., both numbers are biquadratic residues relative to each other taken as moduli. It could be noted that these three cases could point to the fact that biquadratic reciprocity can occur between pairs of 8n+5 numbers, and pairs of 8n+1 numbers, but, as this only occurs in a few places, it would be useless to make such a statement.
It is curious that 73 and 89 are the only 8n+1 numbers for which biquadratic reciprocity appears to hold, and they were also the only moduli for which 2 was a biquadratic residue-perhaps we may find something useful here later, but for now, it holds no weight. Remember, this table is only for 4n+1 primes under 100; the properties we note inductively here, may or may not hold for higher primes.1
With this last observation concerning biquadratic reciprocity, we have stumbled upon a quite important fact. What is the barrier which we have reached, that, although the law of quadratic reciprocity held for all numbers 4n+3 and 4n+1, as we make the next division of 8n+1 and 8n+5, biquadratic reciprocity breaks down completely?
In this section, we have been investigating the tables of real numbers, with the subsuming intention of applying what we can note here of the distinction of 8n+1 and 8n+5, to those 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.
Picture of the same
Now, as we have just recently stated,the number 5, for instance, relative to modulus 61, is a quadratic residue, and 61 is a biquadratic residue of modulus 5. But, think about what we've shown a number of times now, that every 4n+1 number is the product of two complex numbers, which, if either is taken as modulus, both are congruent to 4n+1. For instance, 5 is congruent to 1+2i, if 1+2i is taken as modulus. But hold on a second here..... 5 is also congruent to 1-2i if 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?
What we have come across here, inductively, demonstrates, that in order to discover whether there is law of biquadratic reciprocity, further experimentation demands that complex numbers be the basis of the investigation! Real numbers have failed to reveal to the nature of the biquadratic domain of numbers, as demonstrated precisely here. In a sense, biquadratic reciprocity only has a chance of becoming clear when we are looking at real prime numbers: remember, 4n+1 `prime' numbers are in fact, not prime.
Did Gauss himself reach this barrier, only to solve it by introducing the complex modulus, which we are investigating here? Or, did Gauss exhaust the investigation of biquadratic residues relative to real numbers to find the precise dividing line between real and complex?
Gauss stated in article one of his first treatise how he came to this realization:
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.
In the second treatise he states:
"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 that the natural source of a general theory was to be sought in an expansion of the field of arithmetic, as we already indicated in article 1.
"Namely, while higher arithmetic dealt only with whole real numbers in the questions heretofore treated, the theorems pertaining to the biquadratic residues only appeared in their entire simplicity and natural beauty, if the field of arithmetic is also extended to the imaginary numbers, so that without limit, numbers of the form a+bi form the object itself, where, as usual, i denotes the imaginary magnitude Ö-1 and the indeterminates a,b denote all whole real numbers between -¥ and +¥. We will call these types of numbers complex whole numbers, indeed, so that the real [numbers] do not oppose the complex numbers, but are rather considered as a special case of them."
Is 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.
Another provoking fact is that the complex modulus seems much more important than the biquadratic character of real numbers; yet, it is the barrier one comes upon in the discovery of their 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?
A question arises here: what is the relationship between what you are proving, and the instrument you are using to prove it? The interesting thing is that the instrument is often more profound than the principle, much more profound as a subsuming characteristic relative to the nature of the investigation itself. For instance, in Kepler's 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.
It is as though the the principles which make up space are unfolded, as the mind, in viewing certain characteristics of number, runs into a boundary. Think about what we said about the relationships of the mind at the beginning of this section: numbers are not fixed concepts describing anything usually representable by sense description. The relationships between numbers expresses a characteristic of numbers. However, those characteristics are not constant relative to either of the particular numbers. Thus, what are numbers? What are numbers if they have no fixed properties, and appear to constantly change? What is the medium that is more important than the numbers, which becomes expressed through these changing relationships of the numbers? We've found that certain expressions are constant as particular numbers change their relation with one another. But here, we've reached a boundary simply 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.
We've already seen that Ö{-1} creates a geometry which is purely relations of relations, where the mapping of each relation creates another geometric space, since each number is complex. The complex moduli mapped onto the plane is a space of relations of relations. Therefore we've created a geometry to measure space, which is capable of representing the paradox we've come across inductively, a space unfolded via numbers which constantly change, a priori, as the only means to uncover the principle projecting the paradox into the domain of real numbers.
What does this tell us about number? In this paradox we've come across in investigating the biquadratic character of real numbers, there is a relation between the discrete measurement of arithmetic and the continuous measurement of geometry. The mind, after conceiving of the concept of number physically, and looking at the discrete concepts of integers, discovers paradoxical relations, purely in the mind, which lead to a concept, such as Ö{-1}, demanding a representation in space, but which turns out to be a truthful principle operating in the universe as a whole. What this proves, is that the mind can unfold the principles of the universe, and that those principles are contained within the creative process of the human mind, acting as a subsuming principle ordering the universe as a whole: the principles governing space, exist within the same universe in which human creativity exists.
Restating this crucial point: within one’s own mind, the relationships of concepts lead to boundaries, which although contained in the mind, are a reflection of the fact that the universe in which the geometry of space and the human mind exist, is one. To wit: these principles of human mentation are then directly applicable, in geometric form, to the external physical universe which reflects the same creative principle.
Leibniz' defense of mind, in his `innate principles' argument against John Locke's attempt to reduce the mind to a sponge of sense perceptions, is brought to light.2
Taking imaginary numbers in the physical sense of 90 degree rotations, we thereby return the numbers to their place in circular action and extension, which is the pythagorean, spherically based concept of measurement. Therefore, the purpose of numbers becomes founded, in the minds measure, as Cusa states, the first unfolding of infinite oneness is number: numbers are 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.
Let us now turn to see the full consequence of the realization of the capability to represent a number capable of a higher degree of physical action: Ö{-1}.


1It 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.
2See Leibniz' New Essay on Human Understanding, Chapter 1
3These subjects are discussed elsewhere on this site.

File translated from TEX by TTH, version 3.80.
On 06 Mar 2008, 18:32.