A Journal for Western Man




    Further Corollaries to Stolyarov's Theorem

G. Stolyarov II

Issue XI- February 14, 2003


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^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, 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 gennadystolyarovii@yahoo.com.

This TRA feature has been edited in accordance with TRA’s Statement of Policy.

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Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.

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.