A Journal for Western Man-- Issue XI
G. Stolyarov II
For sake of swifter derivation of fundamental Pythagorean triple relationships, I have devised several new algorithms to assist in this purpose, as well as to supplement the discovery of the triples' periodicity and associations with radial numbers of inscribed circles.
Stolyarov’s Sixth Corollary: Relationships Between Values of Fundamental Pythagorean Triples:
In the Stolyarov Progression, the algorithms for deriving the fundamental Pythagorean triple with radial number R are the following:
a= 2R+1
b= 2R^2+2R
c= 2R^2+2R+1
a^2= (2R+1)^2= 4R^2+4R+1. Also, b+c= 2R^2+2R+2R^2+2R+1= 4R^2+4R+1. Hence, for all fundamental Pythagorean triples (FPTs) of the Stolyarov Progression, b+c=a^2. Since c=b+1, continuing on, we can formulate this in terms of two variables. 2b+1= a^2.
In the Edelman Progression, the algorithms for deriving the fundamental Pythagorean triple with radial number “a” are the following:
x=2a+2
y=a^2+2a
z=a^2+2a+2
x^2= (2a+2)^2= 4a^2+8a+4. Also, y+z= a^2+2a+a^2+2a+2= 2a^2+4a+4, which is half of 4a^2+8a+4. Hence, for all fundamental Pythagorean triples (FPTs) of the Edelman Progression, y+z=(x^2)/2. Since z=y+2, continuing on, we can formulate this in terms of two variables. 2y+2= (x^2)/2.
This algorithm is useful when one knows a single number in a given FPT and is not aware of the others. Instead of performing cumbersome and time-consuming squaring for three numbers, one squared figure is all one needs to obtain; the rest is manageable with the FPT numbers to the first power. The algorithms can also be arranged in terms of the hypotenuse, but conversions therefrom to the longer leg are a matter of elementary subtraction, and shall be left to the reader’s common sense to undertake.
Stolyarov’s Seventh Corollary: A Swifter Method of Calculating FPT Values Using Radial Numbers:
In the Stolyarov Progression, note that a=2R+1 and b=2R^2+2R= R(2R+2). Substituting a for 2R+1, one derives that b=R(a+1).
In the Edelman Progression, note that x=2a+2 and y=a^2+2a= (2a+2-a)(a). Substituting x for 2a+2, one derives that y=(x-a)(a).
This is useful when knowing but the radial number and a single number of a particular FPT. It will swiftly yield the others without extensive variable manipulation. Another convenient rearrangement of these algorithms is, for the Stolyarov Progression, a= b/R-1, and, for the Edelman Progression, x= y/a+a. Derivations for c and z are elementary, knowing their relationships with b and y.
The discovery of these properties once again illustrates the complex beauty of mathematics, and its delightful fathomability of every element in the intricately logical structure of the universe.
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.
