Start with the book Mathematical Physics: Classical Mechanics

But I also want to use the example I am looking for as a guide to direct me in my studies. For example, one of my goals is to learn all the steps necessary to prove Fermat's Last Theorem. I want to do the same thing with the example I requested!

I am starting to study arithmetic geometry and I want to be able to answer the following question asked by a lay person: "what can you do with this thing called arithmetic geometry?" I cited a "child" only to exemplify a lay person who has studied minimum of mathematics. Examples of people to whom I would show the example I asked: my sister, parents, and others I know.

(...) For this reason I would like an example of a Diophantine equation that has a solution consisting of positive integers (because these numbers are intuitive) and that to prove that such a solution exists requires relatively recent results in arithmetic geometry.

My goal is to show to a child (familiar with addition, multiplication, division and exponentiation) mathematical truths that can only be proved with modern techniques of arithmetic geometry. In my opinion Diophantine equations are simple to understand (e.g. $x^n+y^n=z^n$). I know of some theorems (some cited by David Loeffler) that prove that a certain Diophantine equation has no solutions (e.g. Fermat's Last Theorem). (...)

@WillSawin Something relatively recent and that is well explained in a textbook.

@LSpice I liked that PDF, however I want more than one reference.

I think the book link.springer.com/book/10.1007/978-3-319-49834-8 may help you!