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Stillwell mentions in his book, Mathematics and its History that:

Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers...

Have people tried to understand what it is about the "topology" (figuratively speaking) of number theory that makes open problems from it notoriously difficult to solve?

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I'm not convinced there's anything very mathematical going on here (though there might be; I'm not even remotely a number theorist). In order for an unsolved problem to be $n$ years old, it has to be from a field people were studying $n$ years ago. This gives number theory an obvious advantage in terms of age of unsolved problems over, say, algebraic topology. –  Micah Feb 3 at 1:26
To add to @Micah's point, there is only one other equally ancient branch of mathematics whose problems have the age advantage: geometry. If restrict the comparison to just those two fields, an alternative way to ask your question might be, "what is it about euclidean geometry that makes it have so few classical unsolved problems?" –  David H Feb 3 at 1:48
@DavidH Are you going to ask that? You should. –  user89 Feb 3 at 1:52
If you compare number theory with some branches of analysis, then number theory is more `chaotic' in nature in the sense that a small change to the problem (e.g. in the exponent of a Diophantine equation) necessitates the use of radically different tools. Sure, the same problem emerges elsewhere in math, too. But you have to dig a lot deeper to encounter it (me thinks). –  Jyrki Lahtonen Feb 3 at 8:17
@DavidH The answer is simple, unlike Number Theory (and almost all other branches of mathematics) Euclidean Geometry turned out to be deductively Complete. Thus all syntactically valid statements in it are either provable or disprovable within it. Formal systems that are "deductively complete" are also called "uninteresting" because of how easy it turns out to be to prove/disprove all statements once you know this. –  RBarryYoung Feb 3 at 16:08

4 Answers 4

up vote 27 down vote accepted

[The question in the body of the OP is a bit different from the question in the title, and seems more interesting to me. So that is the question I am answering.]

It is common to focus on the fact that it is easy to state number-theoretical problems, since the basic concepts of number theory are fairly elementary. This is certainly true, but it I think it is worth noting another aspect of number theory which can make its problems difficult, which is that the elementary nature of the phrasing of certain problems can belie the hidden structures that underly their deeper nature and (eventually) their solution.

E.g. consider the problem of writing primes in the form $x^2 + n y^2$. When $n = 1$, there is an obvious congruence obstruction mod $4$ to solving $p = x^2 + y^2$, which implies that if $p$ is odd, then in fact $p \equiv 1 \bmod 4$. Fermat showed that this necessary condition is also sufficient.

Similarly, if we want to write $p = x^2 + 5y^2$, then if $p \neq 2, 5$ there is are obvious obstructions: $p$ should be a square mod $5$, and should be a sum of squares mod $4$, and hence $1 \bmod 4$. Together these imply that $p \equiv 1$ or $9 \bmod 20$. Again, Fermat (I believe) proved that this necessary condition is sufficient.

Now consider writing $p = x^2 + 23y^2$. There is no obstruction mod $4$, so the only obvious obstruction is that $p$ should be a square mod $23$. However, this necessary condition is no longer sufficient. E.g. $p = 13$ is a square mod $23$, but can't be written in the form $x^2 + 23y^2$, and the same is true for $p = 29$. (In fact $p = 59$ is the first prime that can be so written.)

Understanding this phenomenon led to many deep discoveries of structure in number theory, such as quadratic reciprocity, and Gauss's theory of quadratic forms. These in turn were major contributors to the development of algebraic number theory and class field theory. From a modern point of view the difference between the case $n = 23$ and the others I considered is that the class group of $\mathbb Z[(1+\sqrt{-23})/2]$ is not a product of cyclic groups of order $2$ --- it is in fact a group of order $3$.

Another example is given by formulas for sums of squares.

There is a formula for the number of ways to write a number as the sum of two squares: $$r_2(n) = 4\sum_{d |n} \chi(d),$$ where $\chi(d) = \pm 1$ if $d \equiv \pm 1 \bmod 4,$ and $0$ if $d$ is even.

There is also a formula for the number of ways to write a number as the sum of four squares: $$r_2(n) = 8 \sum_{d | n, 4\not\mid n} d.$$

There are similar formulas for sums of six or eight squares, but not for higher numbers of squares. It is hard to understand this without realizing that these formulas are governed by the structure of certain spaces of modular forms, which contain no cuspforms in low weights, but which eventually contain cuspforms, for which no elementary formulas are available.

There is an enormous amount of structure present in number theory, despite the elementary manner in which some its problems can be phrased. I believe this is one of the reasons its problems can be so hard to solve; because their phrasing often doesn't make this structure apparent, but finding it is necessary to solve the problems. (The Riemann hypothesis can be thought of as one current illustration of this difficulty.)

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Matt, thanks for the really insightful answer. I am really curious about this notion of "structure" that mathematicians discuss. I have had some glimpses of it (for e.g. in basic number theory problems, or in the "neatness" of geometric (Grassman/Clifford) algebra). I had a question about it here in fact: math.stackexchange.com/questions/660975/… –  user89 Feb 3 at 4:17
...I am just wondering, which field of mathematics would be most appropriate for exploring this "structure" and building intuition about it? Sometimes, I feel that it might be topology, but at other times I am not sure if the answer is just "all of math"...I'd be curious to hear your insight on this. –  user89 Feb 3 at 4:19
@twirlobite: Dear twirlobite, I just saw that question, actually. As for exploring structure, it depends on how serious you are, what you know now, and where you want to go. Maybe you could look at Serre's book A course in arithmetic. It has a lot of fantastic number theory in it, presented in a fairly structural way. It is one of the first books I have my students read, if they haven't already studied it. Another book is Ireland and Rosen's A classical introduction to modern number theory (which is also something I frequently recommend to my students). Regards, –  Matt E Feb 3 at 4:23
@MattE Professor Emerton, I'd heard the comment from your first example before (the class group influencing whether or not a prime is representable by the quadratic form you gave), but not the latter (existence of a formula for sums of squares related to the existence of cuspforms). Is there any good source to read about this relationship? Thanks. –  Mike Miller Feb 3 at 9:03
@Mike: Dear Mike, You could look at the final chapter of A course in arithmetic. Serre doesn't treat the case of sums of squares (because he only handles modular forms for $\mathrm{SL}_2(\mathbb Z)$, i.e. level one modular forms, and you need level four modular forms for sums of squares), but he treats other quadratic forms, and the idea is the same. Regards, –  Matt E Feb 3 at 12:30

I don't think the questions of number theory are necessarily more difficult to answer. Rather, they're easier to ask. Integers are familiar objects and they are easy to play with, and for this reason more questions have likely been asked about them throughout history than about any other objects in mathematics. Once in a while, one of these problems resists all attempts...

That being said, some problems in number theory are incredibly difficult to solve, but incredibly easy to state. How many solutions in integers does $x^{17}+y^{17} = z^{17}$ have? Given a mathematical question, one could try to quantify the ratio of "how difficult it is to solve" to "how difficult it is to state" (perhaps making use of the length of the shortest proof in a given formal language). I wouldn't be surprised if most of the mathematical questions which maximized this ratio in a suitable sense were elementary questions about the integers. Why? I guess it's kind of a philosophical question. Perhaps the integers are the first "highly nontrivial" objects in mathematics, should the objects of mathematics be ordered according to the length of their definition. But really, I have no idea.

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Moreover, problems in Number Theory also get more public attention or general fame: it is much easier for the layman to understand Goldbach's Conjecture than the Navier-Stokes Problem. The publicity received by these problems that are easier to understand can contribute to a false perception of the 'topology' (figuratively speaking, like OP) of open problems. –  Newb Feb 3 at 1:35
@Newb That might be true, although I picked that quote from a text that's not aimed at laymen. –  user89 Feb 3 at 1:36

I had a math professor (elementary number theory) who jokingly hypothesized this was because we live in a continuous world, so proofs are more obvious in analysis than number theory. "Intermediate value theorem? It's continuous function! You drew it without lifting the pencil off the paper. Who thought up that?"

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As DavidH indicated in a comment, we probably know why number theory is able to produce an unbounded number of open problems. As nobody else yet tried to explain this, I will try to give some indications. (If somebody else takes the pain to explain this better, I will gladly remove this answer.) I think it is related to the solution of Hilbert's tenth problem, but already the older incompleteness theorem by Gödel gives a deep reason.

I think the important point about the ("definable" subsets of the) natural numbers is that they allow to emulate the operation of a cartesian closed category. Finding a pairing function is pretty easy, and Cantor already described some simple pairing functions (and their inverses). While the pairing function(s) provide a mean to emulate the product $X\times Y$ of two objects $X$ and $Y$ (note that $X\times Y$ is just another object, and all objects are "definable" subsets of natural numbers in this context), emulating the exponential $Z^Y$ of two objects is much less obvious. It has to describe the "definable" functions from $Y$ to $Z$. (Actually I'm not completely sure what it really has to describe. It must describe the "arrows" between the "definable" subset $Y$ and the "definable" subset $Z$, and I just guessed that these are the "definable" functions between $Y$ and $Z$.) The interesting thing is that "Yuri Matiyasevich utilized a method involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially.", i.e. exponentiation occurs here in two different but related meanings.

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