# A Diophantine Equation and the Modular Method

**Specialeforsvar:** Bertram Koch-Larsen

**Title:** A Diophantine Equation and the Modular Method

**Abstract: **In this thesis, we use the methodologies of modular elliptic curves in order to study the Diophantine equation $a^p+2^\alpha b^p+c^p=0$. In our main theorem 4.4, we show that the equation has no non-zero solutions (a,b,c) when p is an odd prime number and we assume either of the conditions $\alpha\geq2$ or $2\mid abc$ with $\alpha=1$.

The proof follows Ken Ribet's article "On the equation $a^p+2^\alpha b^p+c^p=0$" where we show that when $(a,b,c)$ is a hypothetical solution, either of the assumptions above would imply the existence of an elliptic curve which is simultaneously modular and non-modular: A contradiction.

Besides the goal of proving our main theorem, this thesis also aims to explore the theory behind modularity of elliptic curves.

In section 1, we introduce much of the necessary theory regarding modularity and supporting topics.

In section 2, we apply the theory to two elliptic curves in order to support the claim that they are modular.

In section 3, we prove two results that we need for the proof our our main theorem, which we complete in section 4

**Vejleder**: Ian Kiming**Censor: ** Tom Høholdt, DTU