On the non-existence of rational points on a family of elliptic curves arising from Fermat’s Last Theorem

Kweku A. Opoku-Agyemang

Working Paper Class 16

This paper proves that any positive integer solution to the equation

A^x+B^y=C^z

,where $x,y,$ and $z$ are all greater than $2$ , must satisfy that $A,B,$ and $C$ have a common prime factor. We use the method of a special case of the modularity theorem for elliptic curves, originating from Andrew Wiles. We proceed in two stages. We first state and prove a main lemma that reduces our problem to showing that a certain elliptic curve has no rational points. The lemma shows that if $A,B$ and $C$ are pairwise coprime, then there exists an elliptic curve $E$ that is modular and has rank at least $1$ , and we then show that this elliptic curve is modular and use this fact to derive a contradiction. The additional result shows that $E$ has no rational points of infinite order, except for the trivial ones. This contradicts the fact that $E$ has rank at least $1$ , and hence implies that $A,B,$ and $C$ cannot be pairwise coprime.

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Opoku-Agyemang, Kweku A. (2023). "On the non-existence of rational points on a family of elliptic curves arising from Fermat’s Last Theorem." Machine Learning X Doing Working Paper Class 16. Machine Learning X Doing.

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