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Sometimes when you want to solve an equation you can just use algebra and rearrange it then you are done. But sometimes no amount of algebra can ever prove the equation, and then you need an idea, here is what I mean by the idea:

  • parity (or modular arithmetic for higher numbers)
  • complex integers.
  • the irrational number
  • a special function
  • a strange curve

A lot of equations can be proved impossible if the left hand side is odd and the right hand side is even, sometimes I had a pair of equations that turned out to be easy if you read it as a real and imaginary equation, I read to solve Pell's equation you need a fraction close to the square roots of $d$ and to solve $ax + by = 1$ you have use the greatest common divisor function. To solve Fermat's Last theorem you need elliptic curves.

I have two questions, Where can I find more examples of this (an equation and then what it is really - which lets you prove it)? and how do you find out what is behind the equation aka. what is the idea (because I only see the + - * ^ and variables..)?

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@All: Can we avoid using the tag 'proof'? I've counted 5 only today. In my opinion, it's very uninformative and should not exist. –  Nuno Dec 5 '10 at 22:38
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I interpret the question as "how do I know how to attack a Diophantine problem". The question is in fact not easy and it is this skill that algebraic number theorists hone. Moreover, the methods that you listed are where the story was about 300 years ago. Today, there are more sophisticated and more general techniques available. If you are more interested in the motivation, you should have a look at this MO question, which somewhat goes in your direction.

Before I start listing the modern methods, let me directly address your questions "what is behind a given equation" and "how do I know what to try?", starting with the latter: you don't. After a while you develop some intuition, but still, the basic way to find an approach that works is to try them all out in the order of decreasing likelihood of success. It is this likelihood that you can estimate better and better, as you become more proficient, but you will never know for sure, until you try an approach. As for the question "what is behind a Diophantine equation", there is no good way of making sense of this question at present. Some people will view the equation as describing a geometric object (see last paragraph of this post), some people will look at it from a "modular angle" (see penultimate paragraph). But at the end of the day, when you are interested in integral or rational solutions, the equation is just that: a Diophantine equation. If you must categorise equations, then to categorise them according to the geometry and topology of the set of complex solutions is probably the most sensible thing to do (see last paragraph).

There are two or three rather broad themes in modern research, where modern means everything from the last 150-odd years.

First, a classical method that you have hinted at but that has much more potential is that often, to understand integer solutions, you are forced to work in a bigger ring. You have listed square roots of integers, but the technique is more general than that. To master it, you need to learn some classical algebraic number theory, as it was developed at the end of the 19th - beginning of the 20th century and that's also where I would recommend you to start reading. Have a look at an introductory book into algebraic number theory, such as the book by Ian Stewart, which I personally quite like.

Another broad theme is the one successfully used by Ribet, Frey, Wiles and several others along the way to prove Fermat's Last Theorem. It is nowadays subsumed under the mysterious bracketing term "modularity". To start understanding, what this is about, you first need to learn about modular forms and elliptic curves. The basic idea is that the Shimura-Tanyiama-Weil conjecture, which was the actual result Wiles proved, relates two seemingly unrelated objects: rational elliptic curves and modular forms. This is extremely useful, because modular forms are extremely well-behaved. The "modularity"-idea of solving Diophantine equations then is to construct an elliptic curve out of a putative solution to your given equation that has such strange properties that it cannot possibly be modular. That would then contradict Wiles's theorem, so there cannot be such a solution. The places to start reading about elliptic curves and modular forms are (after you have completely read an introductory book on algebraic number theory and done all the exercises) Silverman - the classic on elliptic curves, and maybe the book by Diamond and Schurman for modular forms.

Finally, a very broad theme is that often, the geometry or the topology of the complex solutions of the equation controls its arithmetic (i.e. Diophantine) behaviour. It is difficult to point to one place where to learn about this, but elliptic curves are definitely the right point to start. I think, once you have read a book on algebraic number theory and one on elliptic curves, you should just come back here and ask this question again with your new background.

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A relatively gentle introduction to modularity is Ash and Gross' pop book "Fearless Symmetry": press.princeton.edu/titles/8141.html –  Qiaochu Yuan Dec 6 '10 at 16:33
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Thanks Qiaochu, I hadn't seen this book. –  Alex B. Dec 7 '10 at 1:37
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It sounds like what you mean is how can I find the motivation for studying a Diophantine equation by looking at the equation. In the case of Fermat's Last Theorem, it is quite a natural equation to study. I would disagree that it is really about elliptic curves, but rather say that it is about the natural numbers. It so happens that elliptic curves gave the tools to solve the problem. Similarly for Pell's equation, you can just wonder about solutions to $x^2-2y^2=1$, find several by hand, and find the recurrence that generates them without knowing about continued fractions and the expansion of $\sqrt{2}$. Knowing about continued fractions and rings where you add a square root of some element suggest new areas of study, which has been quite productive. But I don't know how to discern that from the equation itself.

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I think the poster was asking about how to find points of attack, rather than why bother. But maybe I misunderstand the question myself. –  Alex B. Dec 6 '10 at 0:16
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@Ross: Why do you think that FLT is "natural to study"? Are you aware that almost all leading number theorists (Gauss, Kummer, etc) viewed FLT as an isolated proposition with no connection to the grand themes of number theory? For historical background see e.g. Leo Corry's recent articles –  Bill Dubuque Dec 6 '10 at 0:24
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For what it's worth, I still think that FLT is an isolated problem. It just so happens, that it prompted mathematicians to invent wonderful methods for solving Diophantine equations. But these methods are more widely applicable and there is nothing special in the Fermat equation from the point of view of the methods. By the way, when most of the world was celebrating the solutions of FLT, most mathematicians were celebrating the proof of the Shimura-Taniyama conjecture, which was the actual big achievement of Wiles and which has much more far-reaching consequences than FLT. –  Alex B. Dec 6 '10 at 0:44
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@Alex: FLT did not play a big role in the development of number theory - that's a romanticized myth propagated by inaccurate popular expositions. For more on this see this MO thread –  Bill Dubuque Dec 6 '10 at 1:37
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Actually, it's funny that now, there is no FLT to capture the imagination of ambitious 11 year olds, while algebraic number theory itself is just as vibrant and exciting an area as ever and possibly more so. Just that it's possibly harder to explain to an 11 year old why. –  Alex B. Dec 6 '10 at 4:37
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