Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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16
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1answer
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About Goldbach's conjecture

let's consider a composite natural number $n$ greater or equal to $4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number $r$ such as $(n-r)$ and ...
82
votes
3answers
8k 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 ...
55
votes
4answers
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 ...
32
votes
5answers
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Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence? More formally, 2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA? (This is mentioned in P. ...
23
votes
2answers
2k views

Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
13
votes
1answer
2k views

Distinct numbers in multiplication table

Consider multiplication table for numbers $1,2,\cdots, n$. How many different numbers are there? That is how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there? I'm interested ...
4
votes
3answers
566 views

Bound the error in estimating a relative totient function

Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is $$\Phi(n)=n-\sum_i\frac n{p_i}\+\sum_{i \lt j}\frac ...
0
votes
1answer
412 views

A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained. "let's consider a composite natural number $n$ greater or equal to $4$. ...
167
votes
7answers
82k 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 ...
53
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9answers
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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 ...
32
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6answers
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Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
29
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4answers
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Is there an “elementary” proof of the infinitude of completely split primes?

Let K be a Galois extension of the rationals with degree n. The Chebotarev Density Theorem guarantees that the rational primes that split completely in K have density 1/n and thus there are infinitely ...
45
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2answers
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) ...
41
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6answers
3k views

Algebraic Attacks on the Odd Perfect Number Problem

The odd perfect number problem likely needs no introduction. Recent progress (where by recent I mean roughly the last two centuries) seems to have focused on providing restrictions on an odd perfect ...
29
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4answers
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Are most cubic plane curves over the rationals elliptic?

%This is a new version of the original question modified in the light of the answers and comments. The word 'most' in the title is ambiguous. The following is one way of making it precise. ...
21
votes
4answers
2k views

Which number fields are monogenic? and related questions

A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From ...
10
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5answers
2k views

Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...
17
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3answers
6k views

Proof of the weak Goldbach Conjecture

What are the main ideas of Harald Helfgott's proof that all odd $n \geq 5$ is the sum of 3 primes?
18
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2answers
1k views

Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
4
votes
1answer
422 views

Is there a connection between the closed forms of these two infinite products?

Take the following two infinite products that have closed forms. Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$ The first product: $$\displaystyle ...
8
votes
0answers
664 views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...
8
votes
2answers
1k views

What is the best known upper bound for the number of twin primes?

A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2\, C\, n/ \ln^2(n)$ where $C$ is the so-called twin prime ...
52
votes
21answers
13k 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.
39
votes
28answers
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Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
33
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9answers
5k views

Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
37
votes
3answers
7k views

Proof that pi is transcendental that doesn't use the infinitude of primes

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...
42
votes
6answers
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: ...
42
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3answers
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, ...
38
votes
2answers
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Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is Möbius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
62
votes
6answers
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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 ...
20
votes
3answers
4k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
30
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1answer
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Mochizuki's proof and Siegel zeros

Granville and Stark (Invent. Math. 139 (2000), 509-523) proved that a uniform version of the abc conjecture for number fields eliminates Siegel zeros for $L$-functions associated with quadratic ...
15
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1answer
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On a Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number ...
27
votes
4answers
2k views

Can the difference of two distinct Fibonacci numbers be a square infinitely often?

Can the difference of two distinct Fibonacci numbers be a square infinitely often? There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and ...
39
votes
4answers
3k views

Has Fermat's Last Theorem per se been used?

There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current ...
30
votes
7answers
4k views

Are some numbers more irrational than others?

Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
9
votes
4answers
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One point in the post of Terence Tao on Ax-Grothendieck theorem

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/ ...
27
votes
4answers
2k views

Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ...
11
votes
4answers
3k views

Are the non trivial zeros of Zeta simple?

Hello, a few years ago, I found on ArXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann Zeta function ...
31
votes
2answers
3k views

What is the state of our ignorance about the normality of pi?

Famously, it is not known whether $\pi$ is a normal number. Indeed, there are far weaker statements that are not known, such as the statement that there are infinitely many 7s in the decimal expansion ...
22
votes
4answers
6k views

Primes P such that ((P-1)/2)!=1 mod P

I was looking at Wilson's theorem: If $P$ is a prime then $(P-1)!\equiv -1\pmod P$. I realized this implies that for primes $P\equiv 3\pmod 4$, that $\left(\frac{P-1}{2}\right)!\equiv \pm1 \pmod P$. ...
21
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4answers
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Why so difficult to prove infinitely many restricted primes?

I wondered whether there were an infinite number of palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...) and quickly discovered that it is unknown (OEIS A117697). ...
20
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4answers
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Does pi contain 1000 consecutive zeroes (in base 10)?

The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at ...
16
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5answers
1k views

Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here: Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
11
votes
2answers
610 views

Frobenius density theorem

As mentioned by @MichaelZieve in his comment re Quadratic residue, Chebotarev's density theorem was preceded by an allegedly much easier theorem of Frobenius (Mike Zieve is certainly not the only one ...
28
votes
2answers
694 views

Difference of j-invariant values and the abc conjecture

I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then \begin{align*} \frac{j(i)}{1728} &= 1 \\ ...
12
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1answer
430 views

primes represented by an indefinite binary quadratic form

Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$ Do there exist ...
12
votes
2answers
537 views

Integral points on varieties

I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties. For example, consider $X = \mathbb{A}_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine ...
21
votes
3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot ...
13
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4answers
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Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? : Is it known that there are infinitely many primes p for which ...