A Journal for Western Man-- Issue XI
G. Stolyarov II
Upon a meticulous review of my theorem, I encountered a fundamental Pythagorean triple (FPT) which did not fit a contention of mine in Stolyarov’s Fifth Corollary, that the hypotenuse of an FPT is always equal to either the longest leg plus one or the longest leg plus two. That triple was 20,21,29, and the difference between the two values (b and c) was eight. Faced with the need to derive algorithms for not two but an infinite number of progressions, I set myself to work on determining the pattern in which b and c differences appeared and yielded FPTs. This mandated an examination of not merely particular progressions but algorithms for the derivation of progressions per se.
Stolyarov’s Eighth Corollary: General Derivations of Triple Values in Any Progression:
Before we can fittingly address the question of macroperiodicity, or the occurrence of b and c differences in a fathomable pattern, let us derive general algorithms for a, b, and c values in any given FPT progression. Let d equal the difference between b and c. Hence, (a+b-(b+d))=2R. a-d=2R, and a=2R+d. a^2+b^2=(b+d)^2=b^2+2bd+d^2. Also, a^2+b^2=(2R+d)^2= 4R^2+4Rd+d^2 After cancellation of values on both sides, 4R^2+4Rd=2bd, and b= (2R^2)/d+2R. Then, c= (2R^2)/d+2R+d.
Stolyarov’s Ninth Corollary: Derivations of d Values:
The 20,21,29 triple possesses a d value of 8 and an R value of (20+21-29)/2=6. For its progression, a=2R+8, b=(R^2)/4+2R, and c= =(R^2)/4+2R+8. Interestingly enough, when 2 is substituted for R, the 5,12,13 triple, inverted on its side appears, in the form of 12,5,13 with a d of 8. This is a similar phenomenon to what we had observed with the original triple of the Edelman Progression. Does it follow, then, that every FPT of the Stolyarov Progression, inverted, can yield a progression of its own? I devised an algorithm for a progression of d=18 (assuming an originating triple of 24, 7, 25, the inverted third radial number triple of the Stolyarov Progression), and obtained a fresh progression which included 48,55,73, and 60,91,109. If every FPT of the Stolyarov Progression were to serve as an originating triple for another progression, then acceptable d values (excluding the obvious d=1) would be the following:
2 (the Edelman Progression), 8, 18, 32, 50, 72, 98, 128, 162, 200, 242, 288, and so forth.
Note that 2= 1^2*2, 8= 2^2*2, 18= 3^2*2. This holds true for all the numbers in the aforementioned progression of d values (let us term this the Stolyarov Macroperiodic Difference Series). Its observed algorithm is d=2n^2, where n is equal to the R of the corresponding Stolyarov Progression originating triple. The following is a proof:
In the Stolyarov Progression a= 2R+1 and c= 2R^2+2R+1, and the difference between a and c is 2R^2, or, as we have chosen to term it for the purpose of clarity of nomenclature in the realm of macroperiodicity, 2n^2. Recall, of course, that originating triples of new progressions are inverted, that is, what was termed a in the Stolyarov Progression is treated for derivation purposes as b.
Yet this is not the complete listing of d values, for we have not yet considered the potential of originating triples from the Edelman Progression. In this case x=2a+2, and z=a^2+2a+2, and the difference between z and x is a^2. Of course, the only acceptable n values here are the a values for the Stolyarov Progression. Hence, the Edelman Macroperiodic Difference Series (excepting d=1): 9, 25, 49, 81, 121, 169, 225, etc., where d=n^2.
One may inquire into the possibility of further d values to be derived from inverting any of the non-originating triples from any of the new progressions (where d does not equal 1 or 2), yet there are no new d values to be found within them, due to circumstances that shall be explained later.
Stolyarov’s Tenth Corollary: The Periodicity of R Values within FPT Progressions as a Function of the Macroperiodicity of Differences:
If 12,5,13 serves as the originating triple of a progression instead of 4,3,5, the radius of the “first” FPT in that progression increases twofold. This is also true in the entire sequence of acceptable radial numbers for that progression. That is, instead of being an arithmetic sequence with the first term equal to three and the difference (between terms in the sequence) equal to two, they follow a sequence where the first term is 6 and the difference of radii is 4. Hence, as a general rule of thumb, in a progression with a Stolyarov Progression FPT of radial number n (we shall also term this the periodic number) as the originating triple, the acceptable radial values are the Edelman Progression radial values multiplied by n. In a progression with an Edelman Progression FPT of periodic number n as the originating triple, the acceptable radial values are the Stolyarov Progression radial values multiplied by n.
Here is the proof:
Recalling the algorithms from Stolyarov’s Eighth Corollary, we now substitute 2n^2 for d in an examination of the FPTs with Stolyarov Progression periodic numbers. a=2R+2n^2, b= (2R^2)/(2n^2)+2R= (R/n)^2+2R, c= (R/n)^2+2R+2n^2. Now pretend that n was multiplied by a certain value h. For the new progression, a=2R+2(n^2)(h^2), b=(R/nh)^2+2R, c= (R/nh)^2+2R+2(nh)^2. Given the original range of R values, this is not the set of algorithms which we derived to be universally applicable in Stolyarov’s Eighth Corollary. Hence, it is a false set, and we cannot adjust n values without correspondingly adjusting R values. If R is multiplied by a factor of h, then a=2Rh+2(nh)^2, b= (Rh/nh)^2+2Rh, and c= (Rh/nh)^2+2Rh+2(nh)^2m, which is perfectly consistent with our algorithms given that Rh is the radial number and nh is the periodic number. The same applies to the algorithms for the Edelman series, which would illustrate the exact principle, only with n^2 in place of 2n^2.
Another verification that needs to be made here is that progressions with Edelman Progression FPTs as the originating triples possess radii that can be derived from the integer sequence of radii of the Stolyarov Progression. Upon flipping the 8,15,17 triple on its side we realize that the difference between the c (or z) value (17) and the new a (or x) value (15) is 2. In the subsequent triple of this d=9 progression (21,20,29), that difference is 8. This difference, which equals the c-a difference series of the corresponding terms in the Stolyarov Progression, exhibits such a reverberation of the Stolyarov Progression merely because we have taken the first Stolyarov Progression FPT, flipped it on its side to generate the Edelman Progression, then taken the first Edelman Progression FPT and flipped it on its side to generate the d=9 progression! One would by common sense expect the c-a differences to return to the Stolyarov Macroperiodic Difference Series which once again manifests itself in another periodic relationship, that present within derived progressions with Edelman Progression FPTs as originating triples.
This, to refer to an earlier claim, is the reason why no further d values other than those found in the Stolyarov and Edelman Macroperiodic Difference Series can exist, for the a-c difference values within triples derived from an FPT of one progression are merely the Macroperiodic Difference Series of the other progression!
Stolyarov’s Eleventh Corollary: On the Exclusion of Certain Radial Numbers from the Acceptable Realm:
When I generated the d=18 and d=9 progressions, I noted an oddity whereby radial number multiples of 9 in both progressions yielded non-fundamental triples (36,24,45 for R=9 in the d=18 progression and 27,36,45 for the same radial number in the d=9 progression). This illustrates a phenomenon to be proven by this corollary, and that is that the radial number of any FPT other than one of the Stolyarov Progression cannot equal d/2 or any multiple thereof.
Recall our algorithms for general derivation of FPTs from Stolyarov’s Eighth Corollary:
a=2R+d
b= (2R^2)/d+2R
c= (2R^2)/d+2R+d
Now, substitute d/2 for R:
a= 2d/2+d= 2d
b= (2d^2)/4d+d= 3d/2
c= 3d/2+d= 5d/2
Given that d/2 is an integer number (which it must be to be an R of an FPT), every one of these three values can be evenly divided into d/2, thus not being fundamental. If the value became xd/2 instead, then, likewise the values would become divisible by d/2 multiplied by a factor of x. Hence the reason why 9, 18, 27, 36, and subsequent multiples of 9 are not suitable R values for deriving FPTs in the d=9 and d=18 progressions.
In the Stolyarov Progression, this does not play a factor, because d/2 is not an integer number (it is 0.5), and because b and b+1 are always relatively prime integers because, by common sense, the two of them could never possess a greatest common factor larger than 1.
However, this principle is an explanation for the exclusion of even-numbered radial values from the Edelman Progression, where d is equal to 2. Is d were substituted for R instead of d/2, then x under such values equals 2d+d= 3d, b equals (2d^2)/d+2d= 4d, and z= y+d= 5d, all three of which values are multiples of d (2) and thus non-fundamental.
***
Although I had been mistaken in proposing via Stolyarov’s Fifth Corollary that the Stolyarov and Edelman Progressions comprise the entirety of FPTs in existence, their derivation and analysis nevertheless served as a launching point for discovering a more expansive and profound periodicity amongst the infinity of progressions which now can be derived and whose pattern of radial numbers is inextricably linked to the patterns observable within the Stolyarov and Edelman Progressions. The periodicity of FPTs in accordance to radii of inscribed circles is indeed a phenomenon present throughout the realm of fundamental Pythagorean triples, although in a far more intricate manner than I had initially imagined. I cannot state with confidence that I will not discover new relationships or new means of triple derivations, for my previous assertions in that regard had proven improper and overly swift in bringing a matter to conclusion while the potential for exploring and discovering ever more astounding relationships remained vast. I am well aware, however, that every recent set of corollaries has yielded a more comprehensive grasp of this beautiful occurrence and has advanced its knowledge by Man many steps closer to the pinnacle of absolute truth.
G. Stolyarov II is a science fiction novelist, independent philosophical essayist, poet, amateur mathematician and composer, contributor to Enter Stage Right, writer for Objective Medicine, and Editor-in-Chief of The Rational Argumentator. He can be contacted at gennadystolyarovii@yahoo.com.
(C) 2003 G. Stolyarov II, All rights reserved.
Stolyarov's Theorem and all related concepts are property of G. Stolyarov II.
