1. *GYM* algorithm for searching for Rational points on Elliptic Curves

and

2. *ABC ansatz/algorithm* for generation of Rational Points on Elliptic Curve.

Both algorithms rely on method of *Triangulation of Elliptic Curve*
introduced by author in his first paper on Elliptic Curves in May 1996.

Also it is demonstrated that *Degeneracy* of Rational Points on Elliptic Curve is a *common fundamental reason* for Quantum Effects like Zeeman effect and Stark effect.

1. 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.

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

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

S^{2} = p(p - a)(p - b)(p - c)

can be used for parametrization of an Elliptic Curve E:

y^{2} = x (x - r_{1}) (x - r_{2})

In *RPEC and GYM algorithm* I applied the Triangulation method to the old problem (due to Pierre de Fermat) of finding Rational points on Elliptic Curves. It was shown that there is simple criteria for existence of Rational points on Elliptic curve.

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

3. In my third article on Elliptic Curves "Elliptic Curve ABC ansatz" I again applied the Triangulation method but also used it with special ansatz * A*B = C* to successfully find NEW simpler and faster algorithm for *generation* of Rational points on Elliptic Curve.

Those 3 articles on Rational Points on Elliptic Curve are now used to establish, in two different ways, a *correspondence* between Hydrogen Atom ground states (i.e. solution to the Schrödinger equation for electron in a central potential) and Rational Points on Triangulated Elliptic Curve.

Therefore, splitting of Hydrogen Atom spectral lines in both magnetic field (Zeeman effect) and electric field (Stark effect) may be considered a consequence of *degeneracy* of Rational Points on Triangulated Elliptic Curve.

Establishing degeneracy of Rational Points on Elliptic Curve as *common fundamental reason* for those Quantum Effects takes us one step closer to the *Theory of Everything (ToE)*, which should one day provide Unified Framework for All knowledge accumulated by humanity ever since our history began!

This crucial observation was first made by author when implementing his new *ABC* algorithm for Rational Points generation in Elliptic Curve Applet version 7.0 released in November of 2017 and fully described in new chapter 5 of 3rd Edition of "Motley String or What Everything is made of" book by George Yury Matveev, published in April 2019.

Download article Hydrogen Atom and Elliptic Curve.

There are some other "unorthodox" ideas in my head (related to Elliptic Curves, Number Theory, PDEs 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, job contract, etc.) will be much appreciated and may lead to further progress in those research projects.

Copyright © 2019 George Yury Matveev All rights reserved.