S2 = p(p - a)(p - b)(p - c).
It is shown that each primitive triangle (one with rational area and sides gcd(a,b,c) = 1)
plays role of generator of infinite number of rational points on the same curve.
Explicit formulas are provided for rational points, discriminant and j-invariant.
To see how triangulation works in practice have a look into this page:
Elliptic Curve Applet
Ideas introduced in this article originate back to 1993-1996 when I was living in Montreal, Canada.
In the summer of 1994 I applied for PhD program in Number Theory in the Department of Mathematics and Statistics of McGill University, and was accepted to Graduate Studies as PhD student.
Reference and recommendation were kindly provided by late professor M.I.Pudovkin (head of Magnetosphere Laboratory at the time) based on my Diploma thesis "String model and computer simulation of Solar Flare" results.
But in spite of the fact that tuition fees were kindly paid by Professor Ram Murty
and university produced my student ID card and all documents required, clerk of
Immigration du Quebec did not grant student visa and I was unable to legally
continue my study. So I just attended seminars and lectures on Number Theory
conducted by CICMA (Centre Interuniversitaire en Calcul Mathematique Algebrique)
in McGill, Concordia and UdeM and used university library and computing lab.
First draft of the manuscript was created in the spring of 1996 when I left Canada,
and started working as IT consultant. It remained on paper until spring 2010 when
I finally had enough time (thanks to recession) to put it in TeX/pdf format.
It is published here largely in its original form except for a few numerical results
produced specifically for the pdf version.
Article Rational Points on Elliptic Curve via Triangulation is an extension of this work. It describes GYM (George Yury Matveev) algorithm for finding Rational points on Elliptic curve based on triangulation and one more ancient formula related to it.
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.
There were some other "unorthodox" ideas I had developed in the first half of 1990s (related to Elliptic Curves, Number Theory, PDE and String Theory).
But (what a surprise!), one has to make a living somehow. So it remains to be seen if God allows me to explore 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 (e.g. research grant, scholarship, contract, etc.) would be much appreciated.
Finally, I'd like to express deep gratitude to ALL my teachers: First of them - my unique and incredible Mother Zoya Ivanovna Matveeva (Yarushkina).
Then teachers in St. Petersburg high school #455: Anastasia Sergeevna Konovalova (algebra, geometry), Nikolai Victorovich Levashov (physics, astronomy).
Later yet, teachers/professors in St. Petersburg State University, Department of Physics: Sergey Boldyrev, Leonid Molotkov, Tatiana Sharova, M.I.Pudovkin, to name a few, as well as all those individuals who supported me in a (rather exciting) journey along my world line.
Copyright © 1996, 2018 George Yury Matveev All rights reserved.