Fruit puzzle and 2-isogeny descent

Giadresco, Naná (2024) Fruit puzzle and 2-isogeny descent. Bachelor's Thesis, Mathematics.

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Abstract

In 2014, Andrew Bremner and Allan Macleod published a paper regarding a cubic representation problem. The problem is about finding positive integer solutions $a,b,c$ to the equation $$ \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=N \quad (\star) $$ where $N$ is a positive integer. This equation has a rational solution $(1,-1,0)$, which allows us to transform it into an elliptic curve over the rationals, and therefore the problem can be translated to finding rational points on an elliptic curve corresponding to positive solutions to $(\star)$. Elliptic curves play an important role in both pure and applied mathematics, and this thesis explores the rich theory of elliptic curves that revolves around solving the cubic representation problem, providing more insightful details to the paper by Bremner and Macleod. We explain the transformation from the projective curve defined by $(\star)$ to an elliptic curve in Weierstrass form. This is done through a change of coordinates that maps the rational point $(1:-1:0)$ to the point at infinity on the elliptic curve. We compute the torsion subgroup of the elliptic curve and show that the points on the torsion subgroup do not give nonzero solutions to the original problem so points of infinite order are needed. This leads to discussing the method of `2-isogeny descent’ used to compute the rank of the elliptic curve. We provide some examples of computing the rank and give some lesser known results about finding solutions to a quartic

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Muller, J.S. and Kilicer, P.
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	30 Jul 2024 10:04
Last Modified:	30 Jul 2024 10:04
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/33757

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