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It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mathematical generalizations thereof, others come from geometry in a variety of ways.

For diophantine equations there seem not to be such a direct link to other areas. I would like to roughly understand why the attention of number theorists concentrates on some kinds of diophantine equations and not on others.

Why an equation such as

$x^2-ny^2=1$

or

$x^3+y^3=z^3$

is (or have been) considered worth studying, and not, say, any other random variant such as (if that specific example is not enough nontrivial for you or if it actually happens to have been studied, feel free to substitute it with your favourite "random" diophantine equation):

$x^3+y^5=z^2$ ? So:

Are there any reasons why certain diophantine equations are worth attention besides the mere approachability (i.e. being neither trivial nor hopelessly difficult to analyze)?

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Choose any ordering of the set of all diophantine equations. Let $f=0$ be the first diophantine equation in the list not worth attention... –  Cam McLeman Oct 18 '10 at 12:30

11 Answers 11

$x^2 - ny^2 = 1$ is interesting for at least two reasons: on the one hand, $x^2 - ny^2$ is a norm from the quadratic field, so the equation has to do with the rather natural question of studying units in real quadratic fields. On the other hand (or, really, on a different finger of the same hand) it is just what you want to study if you are interested in rational approximations to square roots of integers, which in some sense are the "simplest" irrational numbers and thus the first context in which you might think about approximating irrationals by rationals.

Similarly, the Fermat and generalized Fermat equations are quite natural in the following sense: there is a long history of studying the interplay between addition and multiplication in integers, and in particular the additive relations between multiplicatively defined sets (primes, perfect powers, etc.) In this context it makes sense to think about $x^n + y^n = z^n$ and things like the Goldbach conjecture. What makes the former more natural? In some sense, it is natural because there's an approach to it! It turns out that the equation $x^n + y^n = z^n$ is intimately related to the geometry of $P^1$ - three points (in some sense the algebraic curve on which all others are based) and to the closely related object X(1), the moduli space of elliptic curves.

There is no hard and fast rule for "which Diophantine questions are interesting" -- but in general it is not so far off to say that the ones which are interesting are the ones where we have at least some idea how to attack them, because the reason we have some idea how to attack them is typically because they're connected to some other mathematical objects of interest.

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And indeed, I'll add -- I never thought the generalized Fermat equations x^p + y^q = z^r were that interesting until I encountered the work of Henri Darmon explaining how these could be connected (at least in some cases) with natural families of abelian varieties with real multiplication over P^1 - 0,1,infty. Until then the problem seemed like a curiosity to me. –  JSE Oct 17 '10 at 1:28
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And the particular example of a ``random'' Diophantine equation $$ x^3+y^5=z^2 $$ happens to be intimately connected to some of the most beautiful invariant theory of the 19th century due to Klein and others! It provides what is, in some sense, the most interesting example of a generalized Fermat equation in the so-called spherical case. Here, there are precisely $27$ parametrized families of coprime solutions, as shown in a very nice paper of Jonny Edwards in Crelle. –  Mike Bennett Oct 18 '10 at 6:27

The question that was asked compares Diophantine equations to differential equations, with the famous differential equations first arising due to physical arguments before taking on a life of their own. The interests of mathematicians long ago in simple questions about geometry or powers of numbers are what gave rise to the classical Diophantine equations. That such equations still have interest is due to them "taking on a life of their own": connections are found with important themes of mainstream mathematics, so those old equations become good examples of advanced theories.

For example, special instances of Pell's equation x^2-dy^2=1 occurred in the work of Greek and Indian mathematicians thousands of years ago. One reason is related to irrationality. Since sqrt(2) is irrational, x^2 - 2y^2 is not 0 when the variables are positive integers and you might ask, particularly in those old days when there was not very advanced math, what the smallest nonzero integral value of x^2-2y^2 could be, and how such values occur. This leads to x^2-2y^2 = +1 or -1, and both equations have many integral solutions by a recursive method, as the Indians knew. If you look at x^2 - 3y^2 = +1 or -1 you find quickly that there's no solution when the right side is -1, so already something new happens.

Another way Pell's equation arises is through questions about polygonal numbers, which were a topic of interest long ago. (I am not going to argue that they have some over-arching signifiance today, but what do you expect people back then to have been thinking about?) Since 36 = 6^2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 is both a square number and a triangular number, you can ask whether there are square-triangular numbers beyond 36, and this is essentially the same as solving an instance of Pell's equation.

Many geometric questions about triangles, esp. right triangles, with integral or rational side lengths lead to low-degree Diophantine equations. The equation a^2 + b^2 = c^2 is too famous to say anything about. Fermat was inspired to show x^4 + y^4 = z^2 has no nontrivial integral solutions in order to prove no right triangle with rational side lengths can have area equal to a perfect square (you can't "square" a rational right triangle). As an unplanned consequence of solving that problem, Fermat had shown the Fermat equation with exponent 4 has no nontrivial integral solutions (replace z with z^2 in the previous equation). The method discovered by Fermat for this problem was his technique of infinite descent, which he was able to use successfully on other problems, including those with a more positive character (i.e., showing some equation has an integral solution, like primes p = 1 mod 4 being a sum of two squares).

The link found later between Pell's equation and unit groups in quadratic rings provided a reason for number theorists to have a permanent conceptual interest in that equation.

Once we get tired of squares all the time, we might look at squares and cubes. The progressions of perfect squares and cubes keep interlacing and you might ask how close they can come (other than the silly case when they coincide, like with 64 = 8^2 = 4^3). This leads to y^2 = x^3 + 1, y^2 = x^3 - 1, y^2 = x^3 + 2, y^2 = x^3 - 2. Already here we see a very different situation compared to Pell's equation, since these equations will have only a finite number of integral solutions; the case of y^2 = x^3 - 2 is a famous example Fermat used to challenge the British mathematicians. We don't know how Fermat showed the only solutions are (3,5) and (3,-5), but Euler discovered later that prime factorization in the ring Z[sqrt(-2)] gives a natural explanation of the result. This was one of the earliest instances of using algebraic integers to solve Diophantine equations in ordinary integers, and still provides a good example for an algebraic number theory course.

Euler looked at y^2 = x^3 + 1 and y^2 = x^3 - 1 and found a way to apply Fermat's idea of infinite descent to show the only integral solutions are the small ones you can find by hand. In the early 20th century Mordell pushed the method of descent further to prove the Mordell part of the Mordell--Weil theorem. Through the influence of Weil and others, the method of descent remains an important tool, although in a language that looks nothing like what Fermat used.

Mordell spent many years of his life studying integral solutions of the equation y^2 = x^3 + k, where k is a fixed nonzero integer. The equation could be justified as having interest because it's one of the simplest examples of an elliptic curve, but it's important for a better reason. The abc-conjecture, which is one of the big open questions in number theory that has connections to many other unsolved problems, does not at first look like it is about Mordell's equation. However, the abc conjecture turns out to be logically equivalent to specific upper bounds on an integral solution (x,y) to Mordell's equation in terms of the parameter k. So, as Barry Mazur once remarked, the Mordell equation is a far more central topic to all of number theory than its rather special appearance suggests.

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The Pell's equation approach to Archimedes' cattle problem is discussed by Lenstra in the Notices fo the AMS (ams.org/notices/200202/fea-lenstra.pdf). –  Chandan Singh Dalawat Oct 18 '10 at 10:43
    
Several elementary problems leading to Pell's equation are in Barbeau's incredibly well-titled book "Pell's equation". The problems lead to a Pell-like equation x^2 - dy^2 = n where d is a small number, usually 2 or 3. –  KConrad Oct 18 '10 at 12:12
    
Rereading this after three years is no less a pleasure than it was originally. –  Mariano Suárez-Alvarez Oct 16 '13 at 23:08

Added 22, October:

While reflecting on this question and the subsequent discussion, I fell again into a bit of a sermonizing mood. I hope I will be forgiven for inserting two words of caution.

  1. There is a difference, probably significant, between lack of interest and aggressive lack of interest. Most people will know what I mean by this, I think. The latter is probably best avoided. Now, I wouldn't be too strong about doing away with it altogether since some people do seem to derive substantial creative energy by being against something, both in mathematics and in the world at large. But for most of us, the less contemplative version of disinterest is, I believe, more destructive than constructive.

  2. Since I've already expounded on the mainstream view, it might be alright to vary on it a bit. It's good to educate oneself about fashion and probably sensible to follow it a fair amount. But then there are the true cliches about independent thinking. I'm old enough to have witnessed first-hand the fervor surrounding the wonderful proof of FLT. At this point, I'm sure many of us can recite by heart all the reasons it is more important than Goldbach, for example, and the significance of the relation to modularity, etc. This may indeed be a reasonable point of view. However, to be honest, I rarely got the impression then that the young expert in the common room was doing more than just that: reciting the viewpoint. I suppose I'm just repeating the platitude that fashion might be quite sensible, but slavish adherence to it is not. So if you feel strongly about some specific Diophantine equation that doesn't quite fit the standing paradigms, my own advice is to to think about it frequently enough to see if some real ideas develop. I hope I'm not misrepresenting him, but Swinnerton-Dyer once claimed that very few people were interested in L-functions around the time he was first experimenting with points on elliptic curves. Even now, he will speak with considerable passion about a single equation, or at least, about a single special family. (Nonetheless, I hope these paragraphs don't strike you as regurgitation of some superficial faith in 'diversity'. I have some of that as well, but some mathematics is clearly better than others.)

Regarding Goldbach, I have the curious impression that it's about to gain substantially in respectability, especially with the remarkable ascent of additive number theory related to the work of Gowers, Green-Tao, et. al. I try to think about it myself every now and then, partly because of the influence of Shinichi Mochizuki, who insists on viewing the connection between additive and multiplicative structures in arithmetic through the prism of non-abelian fundamental groups.


Oops! I'd better clarify right away that my remarks on Fermat and Goldbach are meant in no way as criticism of Jordan's nice answer.


Original answer:

A proper answer might require at least an essay, but here is an abridged attempt.

Two classes of equations have already been discussed in the other answers:

(1) Some equations are 'just interesting' for their special or exotic properties. I quite like the classical mathematics generated by the equation $$x^3+y^5=z^2$$ mentioned in Mike Bennett's comment. Smooth cubic surfaces like $$x^3+y^3+z^3+w^3=0$$ are also nice with their twenty-seven lines that eventually do influence their arithmetic. Fermat equations might be appreciated for their similarly high degree of symmetry that induces the complex multiplication on their Jacobians. By the way, regardless of their age, I find Calabi-yau varieties quite fascinating myself, since the interplay between Hodge-theoretic and Diophantine properties is a subtle phenomenon deserving of study.

There is obviously no objective criterion being offered here, but still an equation may appear as especially interesting in the same way certain spaces are interesting or some animals are interesting. They excite a certain desire to know about them in considerable detail. Investigation with affection is usually richly rewarded in these cases.

(2) Equations that come up while studying some other problem. Pell's equations were mentioned above, and one could consider other norm equations while studying number fields. Keith mentioned also the relation between the far-reaching ABC conjecture and Mordell's equation. One other nice class of elliptic curves of this nature are $$y^2=x^3-n^2x,$$ which were famously connected to the congruence number problem on the area of right angle triangles with rational sides and areas. A spectacular example in the ABC vein is Mazur's study of points on modular curves that gave rise to uniform bounds on the torsion subgroup of all elliptic curves over $\mathbb{Q}$. And then, integral points on Siegel moduli spaces were bounded by Faltings in his proof of the Mordell conjecture. In short, one can even apply one kind of equation to the study of another (family). In any case, for this class, one presumes the equations are as interesting as the motivating problem.

However, the perspective I actually wished to mention sidesteps the question somewhat. The view is perhaps the most mainstream and reactionary possible in this context, but closest to mathematical practice as I see it. It says most equations have or lack interest not in and of themselves. Rather, the main issue is the questions we ask about them. I will remind you of three examples:

(i) Consider the various conjectures on $L$-functions. Say the conjectures of Birch and Swinnerton-Dyer. To oversimplify the case a bit, suppose you could prove the conjecture in full up to the last detail for the single elliptic curve $$498208y^2=x^3+309208472x^2+1204948278x+3920984$$ or with some other choice of coefficients as random as you want. There will be little disagreement that this would be highly interesting.

In case you think that elliptic curves are already deserving of special attention, choose a random collection of homogeneous equations $$f_1=0, f_2=0, \ldots, f_n=0$$ in $m$ variables. Most of the time, they will define a smooth projective variety $X$. Bloch and Beilinson have conjectured that the order of vanishing of $L(H^{2i-1}(X),s)$ at $s=i$ is equal to the rank of the Chow group of algebraic cycles of codimension $i$ homologically equivalent to zero. Being able to prove that statement for any given $X$ chosen randomly would be highly interesting. Of course because one expects this to be so difficult, people concentrate rather on special $X$'s. [In case you are wondering about their relevance, algebraic cycles on $X$ should rightly be thought of as 'generalized solutions'.]

(ii) Continuing with the same notation, suppose $X$ happens to be a Fano variety, which will often happen if the degrees of the polynomials add up to something rather small compared to the number of variables. In that case, Manin has conjectured that the rational solutions in some fixed number field $F$ (depending on the equations) will be Zariski dense. That is, this is a set of equations with very simple geometry, because of which it should be consistently easy to find many, many solutions, as soon as some obvious obstruction is overcome. Once again, you are free to attempt this after choosing the $f_i$'s in as arbitrary and as unaesthetic a manner as possible. As with the Beilinson-Bloch conjecture, the more random your choice is, the more impressive your result will be, in some sense.

(iii) Close to my own heart is the effective Mordell conjecture, which asks for an algorithm to find all rational solutions to a generic equation $$f(x,y)=0$$ with degree at least 4. As a consequence of the fact that such an algorithm is unknown, it becomes of considerable interest to be able to list full solution sets in any given case. Sometimes, it's easy to show that there are none, such as $$x^4+y^4=-1,$$ just to be absurdly simple. However, once such silly reasons for triviality are excluded, for example, if you happen to notice already one solution, it is notoriously difficult to list the whole solution set. Here is an example due to Bjorn Poonen: $$y^2 = x^6 - 2x^4 + 2x^3 + 5x^2 + 2x + 1.$$ You will easily see the solutions $(0,\pm 1)$. However, it requires quite a bit of technology to show that $$(0,\pm 1), (-1,\pm 1), (1,\pm 3)$$ is the full solution set. You can see that this particular equation does seem pretty random. On other hand, because of an enduring focus on the difficult and structurally demanding question of an algorithm, any example of this sort generates quite a bit of interest.

Many people have suggested that the variety of techniques that come up in attacking a problem are as important as the problem itself. There is something to this, in as much as we would like the problem to tell us as much as possible about the mathematical landscape in general, which is, after all, the ultimate object of our investigation. On the other hand, once certain overarching questions have already been established as powerful probes for this process, being able resolve them for any specific object is interesting regardless of how pretty or ugly someone may find the object on its own. Obviously, this is the raison d'etre for good conjectures.


Added 26, October:

Eventually, I stumbled on to the 'box equation' I referred to in the comments. It is $$a_1^2+a_2^2=b_3^2;$$ $$a_1^2+a_3^2=b_2^2;$$ $$a_2^2+a_3^2=b_1^2;$$ $$a_1^2+a_2^2+a_3^2=c^2;$$ defining a surface in $\mathbb{P}^6$. A rational solution with $a_1a_2a_3\neq 0$ corresponds to a 'rational box' having all edges, face diagonals, and long diagonal rational. Apparently, the existence of such a thing is still unknown. There is a nice discussion in this paper of Stoll and Testa. Of course, you have to decide for yourself if it's interesting. The flavor of it is somewhat reminiscent of the congruent number problem, and I think that was why it caught my attention. That is, given my own bias, I had to consider for a minute or two if there were a sneaky connection to a 'conceptually sophisticated' problem. Stoll and Testa relate it, in fact, to the Bombieri-Lang conjecture.

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Dear Minhyong, I like the preliminary discussion that you have added. Perhaps it's reasonable to distinguish "interesting to oneself" and "interesting to many others at this moment in time". If one's taste or intuition suggests that studying a particular equation is important or interesting, then it may well make sense to study that equation. But I think it is true that, at the present time, there aren't any particular Diophantine equations that are valued for their own sake (i.e. as a problem to be solved in their own right), in the way that Fermat was before it was solved. (And even ... –  Emerton Oct 22 '10 at 17:35
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... Fermat got a lot of its cachet because of the way it was entwined with the entire history of algebraic number theory.) Incidentally, has Goldbach ever been unfashionable or unrespectable? It seems to be entwined with the history of analytic number theory in a way at least somewhat analogous to Fermat's connection with algebraic number theory. –  Emerton Oct 22 '10 at 17:38
    
Dear Matt, Thanks for your comments. I can't think of a good equation off-hand either, but I hear of nice ones once in a while and then forget about them. Michael Stoll told me about an amusing one that was called the 'box equation,' I think. Maybe someone knowledgeable can remind me what it was. I'm probably just revealing my own ignorance with the comments on Goldbach. But I did sneakily prepare myself for your objection by writing that it is likely to gain in respectability. –  Minhyong Kim Oct 23 '10 at 0:34
    
Dear Minhyong, thank you for this wonderful answer. I found what you added on Oct 22 particularly inspiring. –  Hailong Dao Oct 25 '10 at 17:19

See Chaitin's paper An Algebraic Equation for the Halting Probability, which is on essentially how to write a Lisp interpreter as a huge Diophantine system. Certainly interesting, but a bit beyond what this MO question is asking for. :)

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Picking up on the theme of the Hilbert problem on diophantine sets: we do know that they comprise all recursively enumerable sets. A diophantine set being only slightly more sophisticated than a given equation (at least at first sight), can we really answer the question "which recursively enumerable sets are interesting?" What that implies, really, is that if you leave the confines of traditional studies (low degree, small number of variables), and diophantine geometry (roughly, equations matching up with famous types of algebraic varieties) the only answer would be "and so what do you find interesting?"

I actually have met a logician who claimed that the theory of large cardinals is not about bogglingly large cardinalities as such, but about certain diophantine equations. Considering the set of theorems in an axiomatic set theory as a recursively enumerable set, this perspective can hardly be refuted. If your mathematical interests are encapsulated in some formal theory, then equally there may be diophantine equations you'd regard as interesting. Of course one hardly expects to deal directly with the equations.

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In part the motivation comes from applications, such as physics. Some of the more recent interest in Calabi-Yau varieties e.g. was triggered by the discovery of mirror symmetry by string theorists. Certain classes of families of diophantine polynomials describe simple types of Calabi-Yau spaces in toric varieties and provide a fairly large number of quite different types of diophantine equations. During the first post-mirror symmetry decade this interest came mostly from classical algebraic geometers, but over the past few years some number theorists have become interested in Calabi-Yau spaces as well. The question of automorphy of Calabi-Yau type motives is an example of a concrete problem. This problem is of interest already in dimension two, i.e. for families of K3 surfaces described e.g. by hypersurfaces in weighted projective spaces or toric varieties. In the case of CY threefolds this problem has played an important role in the work of Clozel, Harris, Shepherd-Barron, Taylor on the Sato-Tate conjecture. This involves a 1-parameter family of quintics in projective 4-space ${\mathbb P}^4$.

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But diophantine equations have been studied for literally millenia! I would say that the role played by the physical motivation is rather minor in the way mathematicians pick "interesting equations"... –  Mariano Suárez-Alvarez Oct 17 '10 at 15:03
    
I of course completely agree that many mathematicians don't pay any attention to physics. But we live now, not millenia ago, and for the past 20 years string theory has had a definite impact on mathematics. It seems to me that as a selection criterion it does occasionally play a role, if perhaps a minor one in the big scheme of things, because there is a lot of interesting structure associated to higher dimensional CY varieties, more so than to some other systems of equations –  Laie Oct 17 '10 at 20:39

My point of view is that one is really interested in the rational points of a particular variety (or class of varieties). The diophantine equation ``comes along for the ride,'' so to speak. For example, one is interested in questions like: does a variety have a rational point? Are the points dense for various topologies (Zariski, adelic)? Does the class of varieties our particular example comes from satisfy the Hasse principle?

It turns out that the answers to these questions tend to be invariant under birational transformations: e.g. if $k$ is a number field, the Lang-Nishimura lemma says that if $X' \to X$ is a birational map between proper integral $k$-varieties then $X'$ has a smooth $k$-point if and only if $X$ has a smooth $k$-point.

This suggests that we let birational classification results help us decide which classes of varieties (and hence what kinds of diophantine equations) to study. Morally, the more geometry we know about a particular birational class, the more we'll be able to say about the arithmetic of the variety (and hence the solutions of the associated diophantine equations), which is to say I strongly agree with JSE's last paragraph.

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As has been well told by others, there are many interesting classes of diopantine equations (norm equations, elliptic curves and abelian varieties, curves of genus > 1, S-unit equations, varieties where the rational points are always Zariski dense or are always finite, failure of the Hasse principle (or not), Lang's conjecture, etc.). However, it is a theorem of Wiles that are no more specific diophantine equations of interest. Any particular problem, say, the existence of infinitely many integral solutions to $x^3+y^3+z^3 = 3$, will only be interesting to the extent that the solution sheds light on the general arithmetic properties of surfaces. Fermat was special, for a combination of historic and aesthetic reasons.

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I think this is a fair assessment of the situation. –  Emerton Oct 19 '10 at 0:08
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Matt: This might be one of those rare instances where we disagree. In any case, I don't understand the mysterious sentence 'it is a theorem of Wiles that [sic] are no more specific diophantine equations of interest.' –  Minhyong Kim Oct 19 '10 at 3:39
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Dear FJ: My quote was certainly not meant as a complaint! Because English is not my first language, when I'm about to make an 'obvious' correction, I'm often grabbed by hesitation. On the other hand, maybe it's that I don't understand the usage of [sic], which I'd be happy to be enlightened about. Yes, there was an obvious reading, but the claim in that form seemed unusually extreme to me, so I wanted to be sure. –  Minhyong Kim Oct 20 '10 at 8:45
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Anyways, assuming the obvious reading, it still seemed a bit too strange for someone to claim that there is a theorem to the effect that something is not interesting. I even wondered if the name 'Wiles' actually referred to someone else, perhaps a logician, who had a theorem that admitted such an interpretation. I guess it was a sophisticated joke of sorts that went over my head. –  Minhyong Kim Oct 20 '10 at 9:01
    
Dear Minhyong and Frictionless Jellyfish, Knowing finiteness of Sha for one elliptic curve of rank > 1 is something that I would certainly find very interesting; it's hard to imagine how this could be verified without the method extending to some larger class of curves (however circumscribed), but in any case, knowing it even in one instance would be fantastic. So perhaps my position is not quite as extreme as curmudgeon's. But I am sympathetic to the idea that Fermat was the last really interesting "individual" Diophantine equation, gaining its interest from itself rather than its ... –  Emerton Oct 20 '10 at 15:30

This certainly isn't something I thought a lot about, but there has definitely been interest about "Generalized Fermat Equations" (like the one you listed). Here's a quick link that I found googling it that has more or less the theorems I remembered: http://www.claymath.org/publications/Arithmetic_Geometry/Chapdelaine.pdf

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The Greeks were interested in geometry but, at the same time, they preferred their quantities to be rational, so they naturally asked when could they construct geometric objects where the measurements were rational. Rational solutions to $x^2+y^2=z^2$ is the obvious example.

Nowadays we are still interested in geometry, albeit of a different kind. Algebraic geometers parametrize geometric objects by moduli spaces and we can still ask when such moduli spaces have points with rational coordinates. The paramount example here is of modular curves. These parametrize elliptic curves with certain properties and having (or not having) one such defined over $\mathbb{Q}$ is nice, so modular curves are interesting diophantine equations.

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Pell equations were used in Matiyasevich's solution of the Hilbert 10th problem. They are related to continued fractions, and many other important things. But originally they, I think, were studied because they are nice looking equations. Cubic equations are related to elliptic curves. Higher genus equations are related to Diophantine approximation and many deep results in arithmetic geometry (Faltings and others). In general, Diophantine equations are considered interesting by themselves (just as physics applications), and if some new method helps solving some class of Diophantine equations, the method is automatically considered useful even though the Diophantine equations may not be useful (yet).

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