As was seen in the last section, we found that by simply squaring every point inside our modulus square, we readily distinguished the biquadratic residues, quadratic residues, and non-residues from one another. We ended by noting a crucial difference in the way the residues were organized into the classes A,B,C, and D for moduli 1+2i and 1+4i, whose norms are 5 and 17. Why did they turn out to have a different characteristic? How do we distinguish these numbers? How do we begin to distinguish numbers in general? Return to the fundamental question: what is the general nature of different types of numbers?
The mind can determine numbers which come in all forms, whether they be triangular, square, or pentagonal, and so on. Quickly one finds the difference between odd, even, prime, followed by the different forms of prime numbers. The true nature of numbers however, as demonstrated in an earlier section, is only seen when numbers are related in cycles with one another, which brings number back to its foundation in astronomical cycles. Once cycles are introduced, then the way in which numbers express themselves through the viewpoint of one particular number, becomes the lens with which to understand and distinguish the true character of different prime numbers. This is where the significance of different types of residues, such as quadratic, cubic, and biquadratic, etc., becomes most clear.
In the last section we just found that the nature of 4n+1 vs. 4n+3 was dependent on the geometrical space created by complex magnitudes related with one another; however, the norms of both complex numbers above are 4n+1 numbers. Therefore, is there a further division of 4n+1 which would yield insight into the characteristic difference which was encountered? We could distinguish the numbers by whether we choose an even or odd value for n. If we choose an even integer, we have 4·2·n+1, in other words 8n+1. On the other hand, if we choose odd integers for n, we have 4·(2n+1)+1, in other words 8n+5.
Looking back at our two numbers whose norms are 5 and 17, we see that these are the two very forms we are speaking of: one is of the form 8n+1 and the other of the form 8n+5.
In approaching the difference in our two numbers 1+2i and 1+4i, we had been investigating the distribution of least residues into their respective classes. However, if we wanted to make a general investigation between the division of complex prime numbers whose norms are 8n+1 an 8n+5, we require a wide array of inductive data. At the least, we would want data for all the complex prime numbers with norms under 100.
But, if we now want to take up the differences between our complex numbers which correspond to norms of 8n+1 versus 8n+5, we face quite a task with our current method. For instance, if we want to conduct an inductive study for all the complex moduli whose norms are under 100, which includes 97, we would have to square complex residues as big as, 9+3i, which would necessitate using over 80 squares on a sheet of grid paper, and covering the whole page in squares! This would be tedious. How could we use what we discovered about numbers that are congruent to the least residues to find a quicker path?
In attempting to organize such a table of least residues, this author stumbled upon quite a notable fact. What we had shown for all complex numbers on the plane includes the special case of complex numbers, a+bi, where b = 0. These special cases of numbers will also be congruent to the least residues in the complex modulus. Inverting this relationship, an interesting fact presents itself. Every one of our least residues will be congruent to one number, 0 through p, where p is the norm, and thus share in all the properties that congruent numbers do. For example in modulus 1+4i, the geometry shows this clearly.
Putting this realization in a table we would have the following:
After noting this curious property we arrive at a extremely useful proposition: every complex number can be investigated through a particular real number, and inversely, every real number through a particular complex number. The usefulness of this remarkable property is that we can now discover the relations of complex numbers by simply using a chart of real numbers! Therefore, to investigate how modulus 1+4i relates to all other numbers, we can use a table of least residues for the number 17 as a guide. Try this out for other moduli and see how the order of the real numbers, which are congruent to the complex least residues, change as the modulus changes.
What we have achieved with this remarkable fact, is that, for a given complex modulus, we have reduced the investigation of an infinite number of complex numbers to the relationship between p-1 real numbers and p-1 complex numbers, in which the truths of their relationship will act as a mirror, to reflect the standpoint with which the complex modulus views the universe of numbers.
Therefore, why not look back to our real number tables from earlier and use them as an easy means for a complete inductive analysis of complex numbers? It seems the tables have turned!
All the quadratic and biquadratic residues for the complex numbers will simply be those congruent to the real numbers in the 2nd and 4th column of the index. The numbers at the top of the columns will be referred to as indices.
For example, modulus 13:
We can now find out the differences between complex numbers which correspond to norms of 8n+1 and those of 8n+5, quite simply, using these real moduli. We now set out to find here, purely inductively in the tables, what we can about these two types of 4n+1 numbers.
(Right Click on the Animation below and select "Zoom In" for better visibility).
Generally, we see a relation between the amount of numbers in a period, and the power which any residue is taken to, that will give one of the numbers in that period: that is, their product equals p-1. For example, all residues in column 4, taken to the 18th power are congruent to 1, and vice versa. Also, all residues in column 2 taken to the 36th power are congruent to 1. Perhaps the reason why all residues are congruent to 1 when taken to the p-1 power is found in this relationship between the factors and the powers. Or, is it the other way around?
Therefore, certain observations from before are made more simple. Looking at the example of these three primes whose norms are of the form 8n+1, it is first seen that numbers having the period [((p-1))/4] are always a biquadratic residues. This now makes more sense because of the just cited relationship of periods and powers. Since any number to the [((p-1))/4] power will always generate a number of period 4, it is easy to see that any of the four numbers in this period 4, taken to the 4 power will equal 1.
Periods in a Period
Let's look more closely now at the residues in a period. How do the periods, which residues have, determine the ordering within a given period? How does the ordering of the residues relate to the factors of a given period? For consistency of terminology, let us designate by `periods in a period', the sequence of periods which reflect the periods, that each residue in the period has. Just as the factors of p-1 determined the different periods, while each corresponded to a given biquadratic character, i.e., either biquadratic, quadratic, or non-residue, so the individual periods are made up of residues which have periods that are factors of the period of which they are a part.
For example, non-quadratic residues, 10, 22, 51, 63 have periods of 8 terms. The period of 8 terms contains residues from the period of 4 terms, and numbers from the period of 2 terms inside of its own period. In this case, the numbers which have a period of 4 terms are quadratic residues, while the numbers having a period of 2 terms are biquadratic residues. As we see here, there are different classes in each sequence, spaced in a particular way.
Take the periods of 3 and 12 terms. 12 is a multiple of 3. From this, every 4th term in the period of 12 terms, will be a residue in the period of 3 terms. Take another example. In the period of 9 terms, every third term is a residue in the period of 3 terms. Since 9 only has 3 as a factor, there must be two residues from the period of 9 terms preceding one of the period of 3 terms. Accordingly, in the period of 18 terms, every second term will be from the period of 9 terms, and the 9th term will be a residue which has a period of 2 terms.
Another example: since 4 has 2 as a factor, every 2nd term in the period of 4 terms will be a residue with the period of 2 terms. Consequently, the sequence of residues in the period of 4 terms will be those which have periods of 4,2,4,1. That is, the second term is a residue which has a period of 2 terms, the first and third terms are residues which have a period of 4 terms.
Moving up to the period of 8 terms: every other residue will be of the `period of periods' of 4 since 4 is a factor of 8. Hence, the sequence of residues in the period of 8 will be those which have the periods of: 8, 4, 8, 2, 8, 4, 8, 1.
Residues can be defined by their relation to different periods of residues of which they are a part.
This also brings us to a unique insight into an ordering principle for the relations between a residue and its indices: given the periods which have a length of terms equal to the prime factors of p-1, all the other periods can be constructed from them, a priori. Every period in modulus 73 will contain residues from the periods 2 and 3 terms. The `periods of periods' which are greater than these two periods, can therefore be constructed a priori.
As seen in constructing the `period of periods' of 4 terms, if one is given the period of 2 terms, the rest of the periods in the `period of periods', are simply filling in the gaps where there is not a factor of 2 terms. Accordingly, in constructing the period of 8 terms, if one is given the `period of periods' of 4 terms, the rest are simply filling in the gaps where there is not a residue from the `period of periods' of 4. In this simple case of 8, since 2·4 = 8, every other residue in the sequence of terms will be of the `period of periods' of 4, and thus, all the rest of the periods in the `period of periods' will be from the period of 8 terms.
Looking closer at these periods, notice that 1 is the collision of all the different sequences contained within a `period of periods'. For example, this happens in the period 36, when a residue from the period 12 is almost at its 12th occurrence and when a residue from the period of 3 is almost at its 3rd occurrence. As every 3rd residue is from the period of 12 terms and every 12th residue is from period of 3 terms, therefore, the 36th number in the period is a residue of 3's 3rd and a residue of 12's 12th. Simultaneously, the period of 9 is almost at 4th occurrence, while the period of 4 is almost at its 9th occurrence.
Inversely, now we know why each modulus will have the particular character of Biquadratic and Quadratic residues for the periods that it does!
Of course, we still only have the numbers they will be according to a specific period. But, although these tables do not show the series of numbers, they do show us, depending on which period the residues have, which class they are. We don't yet have a means to determine a priori, the residues themselves which will make up the periods. That would require a different experiment, going beyond this particular investigation.
If we were to follow through in the former manner, with the rest of our periods, keeping in mind, that any number found in an index that is prime relative to p-1, will have a period of p-1, i.e., is primitive root, we would generate the following for the first 36 residues:
Each of the numbers in this table represent the period which the residue which would be in this position by its period.
Another question to be answered is, why is it that the specific numbers relative to 3+8i, become the residues they are? For example, looking at the table, why is 3 a quadratic residue of 3+8i?
How to find i and -i:
From what we know in earlier investigations about the relations of powers and periods, any residue taken to a power which is a factor of p-1 will be congruent to a number which is of the period such that, the product of that power and the period will equal p-1. The numbers i and -i: for modulus 73, the numbers 27 and 46, i and -i, have a period of 4 terms. Will this be the case in every modulus? Thinking about the nature of the period of i provides an easy answer this question. Therefore to find the period of i and -i, one needs only to take any number to the power of [(p-1)/4] and one of the four numbers of the period belonging to i and -i presents itself. Furthermore, any biquadratic residue of [(p-1)/4], or its factors, taken to the 4th power will be congruent to 1, quadratic residues congruent to -1, while any non-quadratic residue or primitive root will be congruent to i or -i.
For purposes of our inductive analysis, the following is a chart of the collected real number values of i and -ifor 4n+1 primes under 100.
8n+1 vs. 8n+5
As we now have established a general principle of the periods of p-1 to investigate the characteristics of complex moduli, we proceed forth to our former question: what are the characteristic differences of complex moduli, distinguished by their norms of the form 8n+1 or 8n+5? We will note what properties are invariant about each form, and if there are differences within each form, note them also. The following tables are the result of following the prior process, looking at moduli using an Excel spread sheet, and following a similar procedure for all the moduli under 100. The reader may want to print out their own moduli sheets and carry out the induction themselves before reviewing the following tables.
First, looking over all of these tables, confirms what we observed about 73, i.e. 3+8i, that in all the tables, the power of any residue always gives one of the numbers of a corresponding period, which, if the number of the period and power are multiplied, always come to p-1. Generally, am º b where b is a residue of period n, then, mn = p-1, where n is the length of the period.
There will be as many types of biquadratic residue periods for each modulus as there are factors of [(p-1)/4], however, the quadratic residues periods will be a consequence of how many factors of[(p-1)/4] there are, and those remaining factors will be quadratic residue periods.
Above, we showed that we could express the order of the classes of each period of residues, and their respective periods, a priori, from the factors of p-1. Now, looking at tables for all the numbers under 100, what would be a general expression for how the periods in the periods arise? Again, by periods in a period, we mean: to which period each least residue in the sequence of a period corresponds.
As an example of what type of process would be necessary to go about this, designating the factors of p-1 as a,b,c,d, etc., while disregarding 1 as a factor, we find the following general expression for mod. 13 and 17.
General Expression for Modulus 13
General Expression for Modulus 17
To complete a generalized expression for any period, regardless of which number we choose, may or may not be possible, but would require more work. Also, here the biquadratic character of the general expression of the periods is not shown. It would also be useful to include the character to obtain more general expressions for the periods of the biquadratic and quadratic residues.
Another notable distinction that we see in the tables, is the spacing of each class into their respective periods. For example, there are 18 biquadratic residues in mod. 73: 6 of them have period of 18 terms, 6 have a period of 9 terms, 2 have periods of 3 and 6 terms, and there is 1 which has a periods 2 and 1 terms. What is the reason for the distribution? In the following table the second column indicates the length of the period, and the third column, indicates the quantity of numbers that have that length of period.
Before reading forward, look at the table and hypothesize for yourself, why there is the specific quantity of numbers in each period that there are.
Many observations could be made inductively, such as, the numbers of primitive roots for each moduli, are always equal to the (p-1) minus the sum of the two largest quadratic periods. For example in modulus 37, there period of primitive roots is 36. The two largest quadratic residue periods are 18 and 6. 36 minus 24 equals 12, the number of primitive roots in modulus 37. In this way, similar observations could be made, which perhaps may lead to new insight on the interrelations of the classes and periods; however, more broadly all we need to do to answer the question this table poses, is to apply what we learned earlier. In the periods we made before a priori, each period was made up of a period of least residues which corresponded to the factors of that period. In period 10(for example in the cases of 41 and 61) which has factors 2 and 5, every 5th number is a number from the period of 2 terms, and every 2nd number is a number from the period of 5 terms. Since the last number in every period is always 1, and since there will be 4 numbers which have a period of 5 terms, and 1 number with a period of 2 terms, thus, 4 numbers will have to be the of the period of 10 terms, in period the 10 terms. In a sense, since the periods are determined by the factors of p-1, the numbers in each period are determined by the periods within each period.
Another investigation worth noting, but not one we will follow through, is: what are the sums of all the primitive roots for a given modulus? What about non-residues, and quadratic, and biquadratic? All the non-residues add up to zero. What about the sums of the numbers in different periods for quadratic and biquadratic residues? How does that change depending on 8n+1 and 8n+5 moduli.
Lastly, in what on the surface appears the simpler question: how does the character of numbers change, relative to modulus 8n+1 vs. 8n+5? Looking over all of the tables shown above, and finding the character of -1, i, -i, 2, 3, 4, for the 4n+1 primes under 100 we construct the following table:
-1 is always a biquadratic residue in 8n+1, and always QR in 8n+5 , and 2 always a QR in 8n+1 and a non-quadratic residue in 8n+5. i and -i are non-quadratic residues in the cases of 8n+5, but in the cases of 8n+1, they are quadratic residues. They are sometimes biquadratic residues, but only in the cases of 8n+1 only when [(p-1)/4] is divisible by 4. On the surface, the number 3 does not appear to have any regular relationship to either of the forms. 4 is always a biquadratic residue of 8n+1, and a quadratic residue of 8n+5. For those relationships which were constant, let's restate them in a table:
Reaping the First Fruits of Induction
What are the characteristic differences between 8n+1 and 8n+5? First off, what do we know so far about prime numbers in general? In 4n+1 numbers -1 was a QR, which was not the case in 4n+3 numbers. We found out earlier, that this is metaphysically demonstrated by the nature of complex numbers, whose complex moduli squares will always contain an i and -i. There is also something notable about their periods. In 4n+1, p-1 is always divisible by 4, which is not the case in 4n+3. Aha! Here we see an intimate connection between i and -i and the period of length 4. i and -i always have periods of 4, and thus, if these numbers are among the least residues of any period of residues, the prime number -1, must have a period commensurate with 4.
Now, in both cases of 8n+1 and 8n+5, p-1 is divisible by 4, however, in the former it is also divisible by 8, where in the latter it is not. We noted that in 8n+1 moduli, i and -i were always quadratic, while in 8n+5 moduli they are NQ. Mightn't these two observations be related? Think about this. Quadratic residue periods were found to be those which are factors of [(p-1)/2]. As this, in 8n+5 moduli, equals 4n+2, it is obviously not divisible by 4. Thus, i and -i are not quadratic residues of 8n+5 moduli. But, in cases of 8n+1, 4n is divisible by 4, and so, i and -i are quadratic residues. In two cases, 17 and 97, i and -iare biquadratic residues. Can you see why this would be the case? What is it about these numbers which makes [(p-1)/4] divisible by 4?
Why is -1 always only a quadratic residue of 8n+5 moduli but not a biquadratic residue. Why is the period of 2 never a factor of [((p-1))/4], but is always of factor of [((p-1))/2]? Are these two questions related? Since -1 is a quadratic residue of both, we just noted above that both 8n and 8n+4 are divisible by a period of 4 terms, and therefore contain i,-i, both of which, when squared are equal 1. But also, from above we know that, since -1 has a period of 2, [(p-1)/2] has to be divisible by 2 for -1 to be a quadratic. Likewise, [(p-1)/4] must be divisible by 2 for -1 to be a biquadratic. This latter characteristic is the case for 8n+1 numbers, as 2n meets our criterion, but not for 8n+5, as 2n+1 does not.
Because of what we observed about i and -i with respect to 8n+1 and 8n+5 norms, generally, i and -i will never be involved in the periods of quadratic residues of 8n+5.
In 8n+1 norms, [(p-1)/4], 2n, is not divisible by 4, unless n is a multiple of 2. For numbers under 100, these numbers are, 17,33,49,65,81,and 97. Hence, only 17 and 97 have the periods of 4 and hence, i and -i are contained within their periods. In these two cases [(p-1)/2], which equals 4n, is divisible by 4, and thus, quadratic residues will always contain the period of 4. 1
In 8n+5 norms, [(p-1)/2] = 4n+2, is not divisible by 4, thus quadratic residues will never contain the period of 4, i.e., i and -i.
Before moving along, let us make a note about the accomplishment we have achieved. Our approach was riddled with apparent connections and relations which had to be uncovered through a continous process of induction and generalization. Each question led to a new problem, whose resolution brought us further, but then, always confronting us with some invariant principle, which itself, continously transformed by its reason, a seeming disorder, into a unity of composition. Gauss himself was struck by this process as noted above. What does the fact that these prime numbers are formed out of the concepts of the mind, tell us about the mind's investigation of the universe, and the power of the mind to investigate its compositions?
Eisenstein writes of this process in his autobiography:
The intense and sole appeal of mathematics, besides its content, has been for me the peculiar way of thinking required for dealing with mathematical problems. For me, the process of discovering and concluding new truths from known ones, the extraordinary clarity and evidence of theorems, and the ingenious ideas, which were foundations for a whole group of theories, all held an irresistible fascination. No other science had such a rich harvest to offer, and such an inexhaustible material with which to exercise mental capacities...Soon I became used to advancing deeper from single theorems into a wider context and understanding theories as a unity. Thus the idea of a mathematical beauty dawned upon me. For mathematical beauty exists in the same way, that aesthetic beauty does. It is a beauty which can only be understood, if one is able, with full enthusiasm to see the complete picture- a whole system of discoveries, all following a main idea and leading to a final result, all linked together, in harmony and ingenuity, to form an organic entirety, just like a painting. There is also a mathematical rhythm or taste, which leads the analysis and then leads the approach and the development accordingly.[emphasis added]2
In a sense, the mind is investigating the mind of the Creator, and the unity of the composition can be heard by him who conducts the truths as they were intended to be by the composer.
1It is worth noting here about this special case of 17 and 97, i.e., numbers of the form 4n+1 where n is a multiple of 4. Gauss proves in Book VII of the Disquisitiones Arithmeticae that the only polygons which are constructable on the plane are prime numbers of the form 22n+1, which incidently, as we see here, are also the only numbers which have i and -i included in all periods of the factors of p-1.