## Periods of PrimesBy Michael Kirsch 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.
have turned!
## InductionWe 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).
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?
## Periods in a Perioda 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.
Inversely, now we know why each modulus will have the particular character of Biquadratic and Quadratic residues for the periods that it does!
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.
Each of the numbers in this table represent the period which the residue which would be in this position by its period. ## 8n+1 vs. 8n+5As 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.
^{m} º b where b is a residue of period n, then, mn = p-1, where n is the length of the period.
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.
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.
## Reaping the First Fruits of Induction^{1}
conducts the truths as they were intended to be by the composer.
## Footnotes:^{1}It 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 2^{2n}+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.
^{2}Eisenstein Autobiography... |