Diophantine equation without elementary solution but with simple non elementary solution Is there some example of an diophantine equation that satisfies:


*

*No solution is known using elementary methods.

*It is simple to solve using non elementary methods (e.g. using number fields).


My goal is to find good motivation to dive into advanced algebra for someone who is used to solve everything using elementary methods, to show that something that is impossible to solve elementary is really easy using advanced techniques. Ideally if the person can try to attack the equation by himself, give up and then recognize the "simple" solution using advanced techniques and understand it (at least the main idea).
It is not a problem to find some equations as such in Number theory textbooks, but usually those are also solvable using elementary methods. And if there is an equation in which I am confident person will not solve it using elementary methods, it is something with quite complicated proof (extreme example would be Fermat's Last Theorem).
Update: For clarity, let's consider elementary to refer to methods known to Euler (or mathematicians at that time generally). As for simple solution using advanced techniques, that is definitely subjective, and I have currently no idea how to define this, but I believe there is some kind of consensus among mathematicians on things that are simple and elegant.

I think a good example is given by the equation $$x^2-6y^2=1$$ This equation has infinitely many integer solutions $(x,y)=(a_n,b_n)$ determined by $$a_n+b_n\sqrt6=(5+2\sqrt6)^n\space\space........ (n\ge1)$$ Here the calculation of the number $5+2\sqrt6$ is obviously "fundamental" and it is not obvious at all that there are infinitely many solutions if this equation is looked from  "elementary".
In general, for any square-free positive integer $d$, the equation (Pell's) $$x^2-y^2\sqrt d=1$$ also has infinitely many integer solutions given by $$a_n+b_n\sqrt d=(a_0+b_0\sqrt d)^n$$ where $a_0+b_0\sqrt d$ is called the fundamental unit of the quadratic field $\mathbb Q(\sqrt d)$
NOTE.- To take into account here that elementary is a relative concept in mathematics. For many people the text above is quite "elementary".
