I am going to answer this question by questioning its premise. On the one hand you are seeking a "simple motivation to study arithmetic geometry", and on the other hand you want an equation where finding a solution requires "the most modern/advanced techniques of arithmetic geometry".

You don't need to use the most modern/advanced techniques in order to have a simple motivation to study a subject. Old examples, if presented in a compelling way, can be highly motivational to students who are *new* to a subject. Someone who is first learning about PDEs, say, is pretty unlikely to object to an interesting classical example because its solution doesn't use the fanciest modern techniques if they don't yet know the classical techniques.

Here is an example related to an old problem in number theory.

**Theorem**. If a rational number is the area of a rational-sided right triangle, then it is the area of *infinitely many* rational-sided right triangles.

It is not even obvious that there should be a second such right triangle when you know there is a first one.

For instance, $6$ is the area of the $(3,4,5)$-right triangle and it is also the area of right triangles with sides $(7/10, 120/7, 1201/70)$ and $(1437599/168140 , 2017680/1437599 , 2094350404801/241717895860)$.

Those additional examples with area $6$ were not found by a brute force search, but by converting rational triples $(a,b,c)$ where $a^2 + b^2 = c^2$ and $(1/2)ab = 6$ into rational points $(x,y)$ on the elliptic curve $y^2 = x^3 - 36x$ where $y \not= 0$, adding points on that curve, and converting the result back into a rational triple $(a,b,c)$. (When $a, b, c$ are positive, we can interpret them as the sides of a right triangle, but the math works even if any of them are negative.)

That bijection between right triangles with a given area and rational points on an elliptic curve is part of the famous congruent number problem. The proof of the theorem above is based on the non-obvious fact that for $n \in \mathbf Z^+$, the only nonzero rational torsion points on the elliptic curve $y^2 = x^3 - n^2x$ are $(n,0)$, $(0,0)$, and $(-n,0)$, so a rational point with $y \not= 0$ has infinite order in the group law on that elliptic curve.

Here's a different example using elliptic curves.

**Example**. We can easily write $9$ as a sum of two cubes in the positive integers: $9 = 1^3 + 2^3$. Because of the way integral cubes spread out, $9$ is not a sum of cubes of positive integers in any other way. But it *is* a sum of cubes of positive rational numbers in infinitely many other ways, with the next simplest example being
$$
9 = \left(\frac{487267171714352336560}{609623835676137297449}\right)^3 + \left(\frac{1243617733990094836481}{609623835676137297449}\right)^3.
$$
That was not found by exhaustive search, but by adding the point $(2,1)$ on the elliptic curve $x^3 + y^3 = 9$ to itself a few times until a rational point was found with positive $x$ and $y$.

Qualitative results can be motivation too, like Faltings' theorem (Mordell's conjecture).

If your intention is to motivate *yourself* and not a class, then it would help if you indicated your own background in number theory.

notexist. In principle, if a solution exists then you can find it by brute-force search, without any advanced techniques. However, in practice, if the smallest solution is very large, then you'll need some theory to find it. That's what some of these examples are illustrating. $\endgroup$