Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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4
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1answer
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Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
16
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4answers
3k views

Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?

This question relates to a discussion on another message board. Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My ...
8
votes
3answers
382 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
4
votes
1answer
217 views

Total ramification $p = \epsilon \pi^n$ implies $F = K(\pi)$ and minimal polynomial of $\pi$ Eisenstein

Before I state my question, let me give the set-up / "what I know". Let $F / K$ be an extension of number fields of degree $n$. Let $v$ be a (discrete) valuation on $K$ and let $$ \mathfrak{O}_{v} ...
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1answer
213 views

On the equation $3a^2-4b^3=7^c$

How does one find all integer solutions to the equation $3a^2-4b^3=7^c$?
4
votes
2answers
288 views

$\Sigma_{m|n} \mu(m)^2/\phi(m) = n/\phi(n)$?

I'm trying to prove $$\sum_{m|n} \mu(m)^2/\phi(m) = n/\phi(n)$$ My first realization was that $\mu(m)^2 = 1$ iff $m=1$ or $m$ is a squareless factor of $n$ and otherwise is 0. Let $\{1,m_1,m_2, ... ...
5
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1answer
575 views

What connections are there between number theory and partial differential equations?

What connections are there between number theory and partial differential equations?
2
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1answer
55 views

Is there a generalisation of the distribution ratio

From the theory of numbers we have the Proposition: If $\mathfrak{a}$ and $\mathfrak{b}$ are mutually prime, then the density of primes congruent to $\mathfrak{b}$ modulo $\mathfrak{a}$ in ...
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vote
1answer
208 views

Modular forms question

Can someone enlighten me with the question in the next page: http://www.physicsforums.com/showthread.php?p=3208664#post3208664 I am asked to find all the modular forms with weight $k$ which don't ...
2
votes
2answers
456 views

Is $\sin^2 n$ bounded away from zero?

Can one find a number $m$ such that $\sin^2 n \geq m > 0 $ for all integers $n$? By continuity of $\sin x$, it is enough to say that $|n - k \pi| \geq m^{\prime} > 0$ for all integers $n,k$. ...
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vote
3answers
257 views

Prove the product of a sum of powers of primes diverges

I need to prove that this is divergent: $$\prod_{\stackrel{p\leq N}{p\text{ prime}}}\left(1 + \frac{1}{p} + \frac{1}{p^2} + \cdots + \frac{1}{p^k} + \cdots\right),$$ where the expression inside of the ...
10
votes
1answer
372 views

What is a Structured Polyhedron?

In my work on lattice point enumeration of polytopes, I stumbled upon the following sequence: \begin{eqnarray} 1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, ... \end{eqnarray} ...
8
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1answer
348 views

Hecke operators on modular forms

Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just ...
3
votes
2answers
118 views

Step in Proof of Lemma in Narkiewicz _Elementary and Analytic Theory of Algebraic Numbers_

I was looking at the proof of Lemma 2.17 in Narkiewicz Elementary and Analytic Theory of Algebraic Numbers but don't understand a step. Let $p$ be a rational prime, $a$ be an algebraic integer of ...
2
votes
1answer
131 views

Bounded denominators for modular forms

I recently saw a conjecture that a modular form is a congruence modular form if and only if it has bounded denominators. I wonder if one direction or the other is already known to be true? EDIT: For ...
10
votes
3answers
499 views

prove that $\lim_{x\to\infty} \pi(x)/x=0$

I think I might have asked this question before, but I can't find it on the site, so I sincerely apologize if I am making a duplicate. But anyway, I have been working on this proof for several weeks ...
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vote
1answer
151 views

What 'special' properties do real quadratic fields have?

Sorry for the vague title... I've proved a number theoretical result for the imaginary quadratic fields (it was already known for the rationals). I think it would be much easier to sell if I could ...
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3answers
3k views

Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer

Let $a,b$ be positive integers. When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square. Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two ...
2
votes
1answer
88 views

Why is the image of a finite group under a nontrivial homomorphism into $\mathbb{C}-\{0\}$ a set of roots of unity?

Let G be a finite group and $\phi$ a homomorphism from G to $\mathbb{C}-\{0\}$ such that $\phi(G)\neq \{1\}$. It is claimed, in a book Im studying, that $\phi(G)=\{\zeta_n^r : 1\leq r \leq n\}$, ...
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0answers
108 views

Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
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4answers
6k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
3
votes
1answer
179 views

Möbius function generating function terms, radius of convergence

At the wikipedia page for the Möbius function http://en.wikipedia.org/wiki/M%C3%B6bius_function there is an expression for the ordinary generating function for the Möbius function. Is it possible to ...
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1answer
335 views

Hard problem Relationship between CDH and Discrete Log

In Number Theory, Given the Diffie Hellman tuple (ga,gb,gab), Will the given info be helpful in finding the discrete log of g ,i.e a or b? Edit: That is can we use the solution of a Computational ...
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2answers
1k views

Every even integer can be expressed as the difference of two primes?

Every even integer can be expressed as the difference of two primes? If so, is there any elementary proof?
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1answer
2k views

Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
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1answer
1k views

Sums of prime powers

You are given positive integers N, m, and k. Is there a way to check if $$\sum_{\stackrel{p\le N}{p\text{ prime}}}p^k\equiv0\pmod m$$ faster than computing the (modular) sum? For concreteness, you ...
4
votes
1answer
104 views

Partitioning polynomials in $\mathbb{Z}[x,y]$ by the primes they represent

Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For ...
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votes
2answers
1k views

Is the area of a circle ever an integer?

Is the area of a circle ever an integer? I was trying to answer someone else's question on yahoo answers today and I got thumbs down from people on my answer and have come here to get a thorough ...
11
votes
1answer
224 views

Numbers satisfying $\binom{n}{k} = m!$

Let $k,m,n\in \mathbb{N}$ where $1 < k < n-1$. Consider the equation $$\binom{n}{k} = m!$$ which can also be equivalently written as $$n!=(n-k)!k!m!$$ The only instances I found are ...
8
votes
4answers
401 views

Calculating the median in the St. Petersburg paradox

I am studying a recreational probability problem (which from the comments here I discovered it has a name and long history). One way to address the paradox created by the problem is to study the ...
6
votes
1answer
255 views

Common terms in general Fibonacci sequences

Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
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1answer
228 views

Understanding a counterexample to Hasse principle

I'd like to understand a step in a counterexample of Reichardt and Lind to the Hasse principle. The example is given by the equation $2y^2=x^4-17z^4$ (1). (1) has no rational solutions ($\neq ...
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votes
3answers
4k views

What is $\gcd(0,a)$, where a is a positive integer?

I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is ...
1
vote
1answer
202 views

contour integration $\zeta(s)\zeta(2s)$ and $x^s/s^(k+1)ds$

I am trying to do contour integration on $\int_c \zeta(s)\zeta(2s) \frac{x^s}{s} ds$ and $\int_c \frac{x^s}{s^ks}ds$ where c is the line segment joining c-iT c+iT I understand the basic theory ...
6
votes
2answers
588 views

distance between consecutive primes (related to Polignac's conjecture)

Is there an elementary(or not) proof that there are at least two consecutive primes which have difference $2n$ for every natural number $n$? i remind that Polignac's conjecture states that there ...
5
votes
1answer
400 views

Sum of two squares proof

Find two pairs of relatively prime positive integers $(a,c)$ so that $a^2 + 5929 = c^2$. Can you find additional pairs with $gcd(a,c) > 1$? What I know: $gcd(a,c) = 1$ implies that there are some ...
3
votes
0answers
195 views

The discrete Fourier transform of a Dirichlet charachter

I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language. I am trying to find the ...
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0answers
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Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
3
votes
1answer
307 views

Solving a System of GCD Equations

Given a pair of positive coprime integers $a_1$ and $a_2$, I'd like to determine the set of positive integers $x$ which satisfy the following system of $\text{gcd}$ equations: \begin{eqnarray} ...
5
votes
1answer
149 views

Solving in positive integers an equation containing exponentials

I stumbled upon this number theory problem while I was solving another problem. Here is the equation: $$3^kn + 3^{k-1} + 2^m(3^{k-1} + 2h) = 2^{m+l}n$$ where $k \geq 3, h,l,m,n\in\mathbb{N}$, $n$ is ...
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2answers
348 views

Help to understand a proof by descent?

I am trying to understand the proof in Carmichaels book Diophantine Analysis but I have got stuck at one point in the proof where $w_1$ and $w_2$ are introduced. The theorem it is proving is that the ...
5
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1answer
339 views

Does every ideal class contains a prime ideal that splits?

Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$. Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that ...
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votes
2answers
145 views

Basic question about natural density

Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between ...
6
votes
2answers
363 views

Relative density of primes under extension

Let $\mathbb{P}_{\mathbb{C}}$ be the set of Gaussian primes and $\mathbb{P}_{\mathbb{N}}$ the set of primes in $\mathbb{N}$. Let $\pi_{\mathbf{C}}(\sqrt{n})$ be the number of Gaussian primes with ...
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2answers
104 views

Estimating the Primarithm

Let's define the primarithm function, $pog : \mathbb{N} \rightarrow \mathbb{N}$, where $pog(n)$ is the largest number of distinct primes that can divide a natural number $k$, $k \leq n$. Does this ...
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0answers
69 views

Solutions of linear equations over Z [duplicate]

Possible Duplicate: Generating integers from a linear combination of integers Consider the linear equation $a_{1}x_{1} + a_{2}x_{2}+ \cdots +a_{n}x_{n}=b$ such that $a_i,b\geq 0$, ...
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votes
5answers
2k views

calculating $a^b \!\mod c$

What is the fastest way (general method) to calculate the quantity $a^b \!\mod c$? For example $a=2205$, $b=23$, $c=4891$.
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4answers
5k views

Is zero positive or negative?

Follow up to this question. Is 0 a positive number?
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0answers
102 views

Feasibility of a cryptography transformation

This is a follow-up of the question: Transformation We are given $$g^{1/(x+m)},$$ (it is not possible to find $\frac{1}{x+m}$ due to the Discrete log problem), can we find a $k$ such that ...
17
votes
2answers
446 views

Sequence of powers of Gaussian integers capturing all positive integers?

Fix a complex number $z=x+iy$ where $x,y\in \mathbf{Z}$ Consider the sequence generated by the powers $$z^0, z^1, z^2, z^3,z^4 \ldots$$ The question is whether it is possible to capture any positive ...