Article "Rational Points on Elliptic Curves via Triangulation and GYM algorithm" is an extension of work started in May 1996 when I first realized that ancient Herons formula for area S of triangle (a, b, c) with semi-perimeter p:

S2 = p(p - a)(p - b)(p - c)

can be used for parametrization of an Elliptic Curve E:

y2 = x (x - r1) (x - r2)

In my first paper on Elliptic Curves Triangulation of Elliptic Curve it was shown that each primitive triangle (one with rational area and rational sides with gcd(a,b,c) = 1) plays role of generator of infinite number of rational points on the same curve. Explicit formulas were provided for rational points, discriminant and j-invariant.

In my second article on Elliptic Curves I apply the Triangulation method to the old problem (due to Pierre de Fermat) of finding Rational points on Elliptic Curves. It is shown that there is simple criteria for existence of Rational points on Elliptic curve.

Also I describe GYM algorithm for finding Rational points based on two ancient formulas related to area of triangle.

Download Rational Points on Elliptic Curves via Triangulation and GYM algorithm.

In my third article on Elliptic Curves ("Elliptic Curve ABC ansatz") I again apply Triangulation method but also use it with special ansatz A*B = C to successfully find NEW simpler and faster algorithm for finding Rational points on Elliptic Curve.

The new algorithm (which we simply call ABC algorithm) relies on the very first and most fundamental result in Number Theory attributed to Greek Mathematician and Philosopher Pythagoras to whom this article is humbly dedicated.

Download Elliptic Curve ABC.

To see how triangulation works in practice you may have a look into this page: Elliptic Curve Applet

There are some other "unorthodox" ideas in my head (related to Elliptic Curves, Number Theory, PDE and String Theory). But (what a surprise!), one has to make living somehow. So it remains to be seen if God allows me to explore and develop those ideas.

Individuals and research organizations interested in those ideas and their practical applications are welcome to contact me using contact email found on my Resume page.

Any form of financial assistance (research grant, scholarship, contract, etc.) will be much appreciated and may lead to further progress in those research projects.

Copyright © 2017, 2018 George Yury Matveev All rights reserved.