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The following question is in particular reference to the previous question by Bjorn Poonen. I guess I won't even need to give this link, Polynomial representing all nonnegative integers, since this is perhaps the most famous question of MO until now.

I found this question interesting and natural as a curiosity. But from the interest of people, there seems to be more in it. (recall the exclamation What a nice problem! by Gil Kalai). May someone explain me why this question particularly interesting from, perhap, a number theorist perspective ? For example,is there some deeper linkage to other results ?

The same query applies to Fermat polygonal number theorem. Is there anything that these theorems reveal us about the deeper structure of integers?

More generally, I hope we can address the question: What is it that makes some diophantine equations interesting, while others are less so?

(The change was suggested by Kevin O'Bryant)

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    $\begingroup$ Tran- I think this question is just a bit too vague. It's fine to ask questions that were raised in thinking about another question, but just "What's so good about question X" isn't really a question with a good answer (in part, because probably people have lots of different reasons for liking a question). $\endgroup$ – Ben Webster Apr 11 '10 at 15:30
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    $\begingroup$ I believe that people do not have to justify their aesthetic preferences, - and if you are curious to know why they liked this problem, asking it in a comment to the original question would seem much more appropriate to me! $\endgroup$ – Vladimir Dotsenko Apr 11 '10 at 15:36
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    $\begingroup$ I think, "What is it that makes some diophantine equations interesting, while others are less so?" is an excellent question for MO. $\endgroup$ – Kevin O'Bryant Apr 11 '10 at 15:49
  • $\begingroup$ I suspect that this question would be much more positively received if it were phrased that way, rather than about a single question (and somewhat by extension at its author). $\endgroup$ – Ben Webster Apr 11 '10 at 15:52
  • $\begingroup$ Thank you all for suggestion. I delete all my previous comment to make some space. $\endgroup$ – abcdxyz Apr 11 '10 at 15:56
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Your question is probably too general, I simply hope that you'd like to learn some personal experience of people who do their research in diophantine equations or who apply the equations to other areas.

Although my starting research (under- and postgraduate level) was the theory of transcendental numbers, quite tied to diophantine equations (especially the ones related to linear forms in logarithms), I did not try to work on these. (Maybe, because of my father's attempts to prove Fermat's Last Theorem.) But at one occasion I was "introduced" to the Erdős--Moser equation $1^k+2^k+\dots+(m−1)^k=m^k$ ($m$ and $k$ positive integers), Applications of pattern-free continued fractions, and was impressed by the beauty and power of the method which Leo Moser used in 1953 to show that no solution (except $1+2=3$) exists with $m\le 10^{10^6}$. (!) Moser's method could not give much more, and I was happy enough to collaborate on some new ideas and computational achievements of nowadays to significantly extend (after more than 50 years) Moser's bound.

Another favourite diophantine equation is Catalan's equation $x^p-y^q=1$ ($p,q>1$) which was rather recently solved by Preda Mihăilescu. He even managed to avoid linear forms in logs (which are far from beauty because of so many technicalities). There are many steps in the proof, treating some special cases, most of them using completely different methods of diophantine analysis and algebraic number theory. But they are just beautiful! For example, there are two different proofs of the nonsolvability of $x^2-y^q=1$ in integers $x>1$, $y>1$ for a prime $q>3$. The original one, due to Ko Chao ([On the diophantine equation $x^2=y^n+1$, $xy\ne0$, Sci. Sinica 14 (1965) 457--460]; also given in Mordell's "Diophantine equations"), uses the law of quadratic reciprocity in a very elegant way. Another one, due to E.Z. Chein [A note on the equation $x^2=y^q+1$, Proc. Amer. Math. Soc. 56 (1976) 83--84], is extremely short and elementary.

Summarizing, I would say that natural criteria for considering some diophantine equations (and ignoring other) are the simplicity of the equation (isn't $x^n+y^n=z^n$ simple?) and the beauty and novelty of methods to solve it. Probably, the usefullness of the equation has to be taken into account as well.

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  • $\begingroup$ Perhaps I am not very familiar with Diophantine equations, but I have some trouble understanding why simplicity would implies importance. Maybe simplicity is justified as a criterion because the simplicity of the question would often promise the beauty of the solution? Or it is because of our belief that any important phenomenon can be represented by a simple question? $\endgroup$ – abcdxyz May 6 '10 at 4:39
  • $\begingroup$ Simplicity alone is not a criterion of importance. Simplicity plus the beauty of solutions or simplicity plus hardness (no known method works) make the equation attractive. I wonder whether attractiveness can be interpreted as importance... Probably not. But don't mathematicians (at least some) count attractive problems as important?! I belong to this camp. $\endgroup$ – Wadim Zudilin May 6 '10 at 5:05
  • $\begingroup$ I see, thank you for the answer. :) $\endgroup$ – abcdxyz May 6 '10 at 5:46
  • $\begingroup$ After reading your original question I've realized that you now replace "interesting" by "important". My answer is definitely in favor of "interesting" (attractive, challenging, ...). $\endgroup$ – Wadim Zudilin May 6 '10 at 5:46
  • $\begingroup$ Yes, I think at somewhere in the the back of my mind, I put being interesting closer to being important (i.e in the sense of giving a structural understanding of something), which has some distance from being simple, nice... But I totally respect the different view point. $\endgroup$ – abcdxyz May 6 '10 at 6:22

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