Is there a simple-to-understand diophantine equation (in the sense that it's easy to explain to a child) that has a positive integer solution, but to prove that such a solution exists and to find it requires the most modern/advanced techniques of arithmetic geometry?
My intention is to find a simple motivation to study arithmetic geometry.
Thank you for your attention!
Let $n$ be a prime number which has remainder 5 or 7 on division by 8. Then there exists a right-angled triangle with integer side lengths whose area is $n$ times a square.
This is a theorem of Paul Monsky from 1990, a special case of the "Congruent Number Problem". (1990 is not so recent, I know, but there are generalisations to non-prime $n$ due to Ye Tian from the last couple of years; and really cutting-edge stuff will, almost by definition, not be "well explained in a textbook" since it takes time for ideas to percolate down from the research frontier into instructional texts.).
That said, it doesn't happen all that often that you can can prove that a Diophantine equation has solutions by any method other than simply writing down a solution. There are some exceptions, such as the above. But if you change the question a bit, and allow proofs of non-existence of solutions, or non-existence of any solutions other than some explicitly described set of "known" ones, then you have an immense range of examples, starting with Wiles' proof of Fermat's Last Theorem.
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.
Which integers are the sum of three cubes?
This question remains unanswered in general. You may have heard of this problem in the popular press because there was recent computational progress, but there is a lot more to this problem than meets the eye. Cubic reciprocity plays an important role. If cubic reciprocity is not modern enough, Brauer–Manin obstructions also come into play. For further reading, see the 2021 paper On a question of Mordell by Booker and Sutherland, which shows that nowadays, even a computational search for solutions to an innocent-looking equation needs to be informed by analytic number theory, elliptic curves, etc.
