A Journal for Western Man
Corollaries to Stolyarov's Theorem
G. Stolyarov II
Issue X- February 7, 2003
Every possible integer radial number can be associated with a fundamental Pythagorean triple consisting of the lengths of the sides of the right triangle into which a circle with a radius of the radial number is inscribed. The relationship between the radial number R and the three numbers of the triple is as follows:
c= 1+ 4k=1RΣk
Further examination of Pythagorean triples as well as the properties of deriving them has yielded methods of solution significantly ameliorated from the original, as well as opened pathways into a previously untapped progression of fundamental Pythagorean triples. The following are several corollaries to my theorem, which further expand the concept of periodicity of fundamental Pythagorean triples.
Stolyarov’s First Corollary: Alternate Derivations of b and c:
The calculation of an arithmetic series is a rather cumbersome means of determining the leg of a triangle, and I pondered whether a method existed, applicable across the board to all fundamental Pythagorean triples, for the determination of b and c.
Consider the property of right triangles that a^2+b^2=c^2. In the case of all fundamental Pythagorean triples we have thus far observed (and it will be noted later that yet others exist), c=b+1, so a^2+b^2=(b+1)^2=b^2+2b+1, and a^2=2b+1. Recall now that a=2R+1. Hence, (2R+1)^2=2b+1. After simplification, 2b=4R^2+4R and b=2R^2+2R. Of course, it follows that c=2R^2+2R+1.
Hence, the updated means of deriving the longer leg and the hypotenuse values for a fundamental Pythagorean triple:
b= 4k=1RΣk =2R2+2R
c= 1+ 4k=1RΣk =2R2+2R+1
Stolyarov’s Second Corollary: The Edelman Progression
It was just when I thought myself to have uncovered the key to every single fundamental Pythagorean triple in existence, that a friend of mine, Mr. Lucas Edelman, had informed me of a fundamental triple, with radial number three, which was not structured in accordance to the derivation pattern elucidated above. That triple, a rather basic one (which I had overlooked as a result of the assumption that c=b+1 for all fundamental Pythagorean triples), was 8,15,17. A meticulous analysis of its makeup guided me to the conclusion that it was but the first link in yet another periodic progression of Pythagorean triples. Herein it will be referred to as the Edelman Progression, as distinguished from the Stolyarov Progression which we had derived earlier.
Take 8,15,17 and refer to its legs as x (shorter) and y (longer), and to the hypotenuse as z: R= (x+y-z)/2= (8+15-17)/2=3. x = 8, which is 2*3+2. Thus, in triples of the Edelman Progression, x=2R+2.
Now, to derive y when z=y+2: x^2+y^2=(y+2)^2=y^2+4y+4. Thus, x^2=(2R+2)^2=4y+4. 4R^2+8R=4y, and y=R^2+2R. Consequently, z=R^2+2R+2.
To summarize, values of fundamental Pythagorean triples in the Edelman Progression are determined by the following algorithms:
Stolyarov’s Third Corollary: Periodic Relationships between the Stolyarov and Edelman Progressions:
Assuming that the first radial number of the Edelman Progression is 3 (which is in some respects misleading, as shall be discussed later), we can devise the following chart of the first several fundamental Pythagorean triples in that progression:
x= shorter leg y= longer leg z= hypotenuse R= radial number
8 15 17 3
10 24 26 4
12 35 37 5
14 48 50 6
As I scanned for patterns in these values, I realized the even-radial-numbered triples were not fundamental! 12,35,37, however, is an entirely new addition to the triples that can be derived, and so is 16,63,65 with radial number 7. Only odd numbers from 3 onward yield new fundamental triples. An odd progression I considered it to be (and it was, although in a different sense), and I yearned to determine a reason for such a structure as well as to synthesize it, along with the derivation method for the Stolyarov Progression, into a universal means of deriving fundamental Pythagorean triples at the will’s command.
Such a pattern I did notice. The radial number for the first new triple in the Edelman Progression is equal to 3, which is also the length of a in the first triple of the Stolyarov Progression. This holds true throughout the Edelman Progression.
Therefore, we can derive an Edelman Progression triple along with a Stolyarov Progression triple from every single radial number in existence, which further proves my contention of the periodicity of fundamental Pythagorean triples. All that must be performed is a modicum of variable manipulations, namely, replacing the Edelman Progression radial number with a.
Hence, for radial number R of a Stolyarov Progression fundamental Pythagorean triple:
b= 4k=1RΣk = 2R^2+2R
c= 1+ 4k=1RΣk =2R^2+2R+1
The shorter leg of the Stolyarov Progression fundamental Pythagorean triple is equal to the radial number of the Edelman Progression fundamental Pythagorean triple.
Stolyarov’s Fourth Corollary: The Common Origin of the Stolyarov and Edelman Progressions:
Aside from the equivalence of a and the Edelman Progression’s R, an interesting observation can be experienced should one conduct a mathematically proper but theoretically flawed operation (as it will lead to an already derived triple) and substitute 1 for R in the Edelman Progression. x=2(1)+2=4, y=1^2+2*1=3. z=1^2+2*1+5. The resultant triple is indeed fundamental. It is, as a matter of fact, the most fundamental one of them all, 3,4,5, only in a warped form, in this case, 4,3,5. The operation was flawed, as the shorter leg was calculated as longer than the longer leg (hence this cannot be considered the first triple in the Edelman Progression), however, it does lead to a profound discovery: the Edelman Progression is derived from the first Stolyarov Progression triple flipped onto its side! Hence, the statement that encompasses Stolyarov’s Fourth Corollary is as follows:
The Stolyarov and Edelman Progressions both originate from the 3,4,5 fundamental Pythagorean triple, and are equal when both of their radial numbers are 1.
Of course, the above statement is merely an answer to the question of why an Edelman Progression would exist in the first place. It is not to be applied for derivation purposes, as a triple of the Edelman Progression cannot, under sound theoretical mathematics, possess a radial number of 1 (that would render the Stolyarov Progression triple non-Pythagorean and consisting of fractional values).
Stolyarov’s Fifth Corollary: The All-Encompassing Nature of the Stolyarov and Edelman Progressions:
Note that in the fundamental triple 3,4,5, c=b+1 under the sound premise that the shorter leg is equal to 3. Under the imaginary and sheerly speculative assumption that the shorter leg is equal to 4, c=b+2. No other value for c is possible, therefore no progressions but the Stolyarov and Edelman Progressions exist. It is from a synthesis of the aforementioned progressions that every fundamental Pythagorean triple can be derived and pegged onto a Stolyarov radial number so that a periodic arrangement can be rendered possible. Presently, the full potential of fundamental Pythagorean triple derivation has been tapped.
My endeavors in regard to the 8,15,17 triple, a “rogue triple”, as I had referred to it during moments of my despair, have demonstrated merely that this universe is structured in accordance with perfect reason, that Natural Law is immutable and randomness can claim no legitimate place in it. There are but two elements in our world, the matter to grasp and the will to grasp it, the latter with limitless potential to fathom and manipulate the former. We, men, are that will, the volitional element of the universe, its dauntless explorers and resolute conquerors.
G. Stolyarov II is a science fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, contributor to Enter Stage Right, Le Quebecois Libre, Rebirth of Reason, and the Ludwig von Mises Institute, Senior Writer for The Liberal Institute, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. His newest science fiction novel is Eden against the Colossus. His latest non-fiction treatise is A Rational Cosmology. Mr. Stolyarov can be contacted at email@example.com.
This TRA feature has been edited in accordance with TRA’s Statement of Policy.
Read Mr. Stolyarov's new comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, at http://www.geocities.com/rational_argumentator/rc.html.