**261**

votes

**4**answers

21k views

### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\ $ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?

**163**

votes

**7**answers

10k views

### Polynomial representing all nonnegative integers

Lagrange proved that every nonnegative integer is a sum of 4 squares.
Gauss proved that every nonnegative integer is a sum of 3 triangular numbers.
Is there a 2-variable polynomial $f(x,y) \in ...

**159**

votes

**7**answers

78k views

### Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...

**88**

votes

**12**answers

12k views

### Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

**73**

votes

**11**answers

14k views

### Knuth's intuition that Goldbach might be unprovable

Knuth's intuition that Goldbach's conjecture (every even number greater than 2 can be written as a sum of two primes) might be one of the statements that can neither be proved nor disproved really ...

**70**

votes

**6**answers

8k views

### Why does the Riemann zeta function have non-trivial zeros?

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...

**69**

votes

**4**answers

25k views

### Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville :
“The big experts in the field had
already tried to make this approach
work,” Granville ...

**62**

votes

**2**answers

6k views

### Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open.
I would think that the question of its convergence is really ...

**61**

votes

**6**answers

6k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**60**

votes

**8**answers

6k views

### Why is the Gamma function shifted from the factorial by 1?

I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...

**57**

votes

**2**answers

3k views

### Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...

**55**

votes

**2**answers

4k views

### Riemann hypothesis via absolute geometry

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over ...

**52**

votes

**21**answers

12k views

### What's the “best” proof of quadratic reciprocity?

For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.

**52**

votes

**9**answers

5k views

### Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points.
I am interested to know why Mordell and ...

**52**

votes

**10**answers

6k views

### “Understanding” Gal(\bar Q/Q)

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers G = Gal(Q bar/Q). What do people mean when they say this? The ...

**52**

votes

**1**answer

3k views

### Is there a “classical” proof of this $j$-value congruence?

Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve ...

**52**

votes

**7**answers

3k views

### Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form.

OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I ...

**51**

votes

**3**answers

2k views

### Perron number distribution

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...

**50**

votes

**35**answers

8k views

### Where is number theory used in the rest of mathematics?

Where is number theory used in the rest of mathematics?
To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to ...

**50**

votes

**6**answers

4k views

### Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...

**49**

votes

**5**answers

18k views

### Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...

**49**

votes

**11**answers

4k views

### Why certain diophantine equations are interesting (and others are not) ?

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" ...

**49**

votes

**1**answer

3k views

### Smooth proper scheme over Z

Does every smooth proper morphism $X \to \operatorname{Spec} \mathbf{Z}$ with $X$ nonempty have a section?
EDIT [Bjorn gave additional information in a comment below, which I am recopying here. -- ...

**49**

votes

**4**answers

6k views

### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...

**47**

votes

**4**answers

5k views

### Etale cohomology — Why study it?

I know (at least I think I know) that some of the main motivating problems in the development of etale cohomology were the Weil conjectures. I'd like to know what other problems one can solve using ...

**47**

votes

**2**answers

3k views

### The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...

**47**

votes

**3**answers

4k views

### Is there a “Basic Number Theory” for elliptic curves?

Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...

**46**

votes

**7**answers

3k views

### How does “modern” number theory contribute to further understanding of $\mathbb{N}$?

I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".
I'm a graduate student with interest in number theory. I ...

**46**

votes

**1**answer

2k views

### A function whose fixed points are the primes

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...

**45**

votes

**5**answers

8k views

### Inaccessible cardinals and Andrew Wiles's proof

In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.
Here's the link:
...

**45**

votes

**3**answers

3k views

### Are there infinitely many integer-valued polynomials dominated by $1.9^n$ on all of $\mathbb{N}$?

The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of ...

**45**

votes

**2**answers

3k views

### Non-abelian class field theory and fundamental groups

Over the years, I've been somewhat in the habit of asking questions in this vein to experts in the Langlands programme.
As is well known, given an algebraic number field $K$, they propose to replace ...

**45**

votes

**3**answers

3k views

### Can a positive binary quadratic form represent 14 consecutive numbers?

NEW CONJECTURE: There is no general upper bound.
Wadim Zudilin suggested that I make this a separate question. This follows
representability of consecutive integers by a binary quadratic form
where ...

**45**

votes

**2**answers

4k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**43**

votes

**9**answers

9k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**43**

votes

**3**answers

5k views

### who fixed the topology on ideles?

I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not ...

**43**

votes

**5**answers

3k views

### Does anyone know a polynomial whose lack of roots can't be proved?

In Ebbinghaus-Flum-Thomas's Introduction to Mathematical Logic, the following assertion is made:
If ZFC is consistent, then one can obtain a polynomial $P(x_1, ..., x_n)$ which has no roots in the ...

**43**

votes

**4**answers

3k views

### Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees ...

**42**

votes

**2**answers

4k views

### Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...

**42**

votes

**3**answers

2k views

### What is the limit of gcd(1! + 2! + … + (n-1)! , n!) ?

Let $s_n = \sum_{i=1}^{n-1} i!$ and let $g_n = \gcd (s_n, n!)$. Then it is easy to see that $g_n$ divides $g_{n+1}$. The first few values of $g_n$, starting at $n=2$ are $1, 3, 3, 3, 9, 9, 9, 9, 9, ...

**42**

votes

**0**answers

3k views

### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here ...

**41**

votes

**12**answers

4k views

### Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So ...

**41**

votes

**8**answers

3k views

### The inverse Galois problem, what is it good for?

Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience ...

**41**

votes

**4**answers

6k views

### Zagier's one-sentence proof of Fermat's theorem.

Zagier has a very short proof ( MR1041893) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \in ...

**41**

votes

**6**answers

2k views

### Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?

The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does.
Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then:
$\Gamma(s)-\Gamma(1-s)$ yields zeros at:
...

**41**

votes

**3**answers

1k views

### What is the geometry of an undecidable diophantine equation?

As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...

**41**

votes

**1**answer

2k views

### Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...

**41**

votes

**0**answers

2k views

### the topology of arithmetic progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...

**40**

votes

**14**answers

4k views

### What is the high-concept explanation on why real numbers are useful in number theory?

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...

**40**

votes

**6**answers

3k views

### What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...