Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

learn more… | top users | synonyms (1)

2
votes
1answer
262 views

Proof regarding Robin's inequality (RI).

Let $\sigma$ be the divisor sum function, $\gamma$ the Euler-Mascheroni constant and $n>5040$. Robin showed that if the inequality$$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$$ ever fails, it ...
1
vote
0answers
34 views

$\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ solve for positive integers. [duplicate]

Solve for positive integers $x,y,z$ $$\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ I tried to solve it by some generel work but it didn't help.
2
votes
2answers
117 views

Can sieve method prove ternary (three) prime Goldbach conjecture?

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ? I had done some research, I could not find any articles on this. Can anyone provide some help on this ? I ...
2
votes
1answer
842 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some prime ...
4
votes
2answers
102 views

Find two positive integers $x$ and $y$ such …

Find two positive integers $x$ and $y$ such $$\sqrt{69+20\sqrt{11}}=\sqrt{x}+\sqrt{y}$$ I have worked intensively with this task but I really can't find a solution to this problem. I hope that I ...
3
votes
3answers
183 views

My (divergent) summation of the zetas with sets of cofactors give systematically errors of simple integer differences. What am I missing?

This is a "fiddling" in a small project of mine with which I'm concerned from time to time for three years now. I try to focus on the core of the problem, please ask if more context is needed. ...
1
vote
0answers
37 views

Number of triples $(a, b, c)$ with $1 \leq a,b,c \leq n$ which are coprime ($gcd(a,b,c)=1$)

Number of ordered triples $(a, b, c)$ with $gcd(a, b, c) = 1$ and $1 \leq a, b, c \leq n$ can be computed using the following formula: $$ C(n) = \sum_{k=1}^n\mu(k) \left \lfloor \frac{n}{k} \right \...
0
votes
2answers
65 views

Show there is no solution to this equation

I have to show that $2x^4-20x+8$ cannot be divided by $16$ without remainder. The only thing comes to my mind is to write $16$ as $4^2$ which hasn't been of any help. Could you give me some hints to ...
1
vote
0answers
27 views

what is the best Schnirelmann Constants?

what is the best Schnirelmann Constant for Goldbach Conjecture ? On http://mathworld.wolfram.com/SchnirelmannConstant.html the best Schnirelmann Constant is 7 ( from Ramaré ) My understanding is ...
11
votes
0answers
254 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything $...
0
votes
1answer
41 views

Is it true that an equivalent 'absolute value' is an absolute value?

I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by $|x|'=|x|^r$...
2
votes
0answers
59 views

Number Theory: Ramification

The question is as follows. Let $K = \mathbb{Q}(\sqrt[m]{a},\sqrt[n]{b}) $, where $m,n,a,b$ are positive integers such that they are pairwise coprime. Assume that $[K:\mathbb{Q}]=mn$/ Prove ...
1
vote
1answer
39 views

Number of integers that are sum of two squares

I know that there's a proof (of Landau from 1908) that the numbers of integers that can be represented as sum of two squares which are smaller than $n$ is $$ \Theta\left(n\over\sqrt{log (n)}\right) $$...
0
votes
2answers
106 views

Ramification and Quadratic Reciprocity Law

I have a question regarding the follow problem: Show that the prime number 27644437 splits completely in $L = \mathbb{Q}(\sqrt{55})$. From what I understand. This deals with ramification....
3
votes
2answers
63 views

Proof of Power of Twos and Threes

Are $(1,2), (2,3), (3,4)$, and $(8,9)$ the only consecutive integers that are a power of two and a power of three? And if they are, how do I prove this?
2
votes
2answers
162 views

Prime elements in the gaussian integers

Prove: If a prime number $p\in \mathbb N$ is from the form $p=4k+3,k\in \mathbb N$, then its also a prime number in $\mathbb Z[i]$,i.e. if $p|(z_1\cdot z_2)$ then $p|z_1$ or $p|z_2$. I dont have any ...
1
vote
0answers
72 views

Rationality of sum of roots (or rational function thereof) of a system of algebraic equations.

I am looking for a reference/hints of proof towards statements of the kind; Given an irreducible system of $n$ polynomial equations over $\mathbb Q$ in $n$ variables $$P_i(x_1,...,x_n)=0,\quad i=1,......
3
votes
1answer
170 views

The frequency occurrence of primes in the unique prime factorization of natural numbers.

I am curious about the frequency of occurance of prime numbers in natural numbers. For example starting with the first non-prime 4 = 2^2, then 6 = 2x3, 8 = $2^3$, 9 = $3^2$ etc. Now of course a prime ...
10
votes
1answer
159 views

Are fractions with prime numerator and denominator dense?

So the question is in the title. And I mean dense in positive real numbers ofcourse. Somehow I cannot grasp if this is very trivial or not. The prime numbers aren't that dense, but are there enough of ...
4
votes
1answer
148 views

Proof of $\sum_{n=1,3,5,\ldots}^{\infty}\frac{1}{n^4}=\frac{\pi^4}{96}$

I came across with the infinite series $$\sum_{n=1,3,5,\ldots}^{\infty} \frac{1}{n^4}= \frac{\pi^4}{96}$$ when calculating a problem about an infinite deep square well in quantum mechanics. ...
0
votes
1answer
46 views

Finding $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{p=1}^{\infty}\frac{1}{mnp(m+n+p+1)}. $

Find $$ \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{p=1}^{\infty}\frac{1}{mnp(m+n+p+1)}. $$ Use $$ \frac{1}{m+n+p+1}=\int_{0}^{1}{{{t}^{m+n+p}}\,dt} $$ so the sum equals $ -\int_0^1\ln^3(1-t)\,dt....
1
vote
1answer
72 views

Method to solve quadratic congruence

I learned quadratic congruence by myself and stuck in these problems: I know if quadratic congruence $X^2=a(\mod\mbox{ p} )$ with $p$ is an odd prime number and $\gcd(a,p)=1$, then it has no ...
1
vote
1answer
76 views

Solving $\frac{x}{y}$ mod $m$ efficiently?

We know that : $( x.y )$ mod $m$ = ( ($x$ mod $m$) . ($y$ mod $m$) ) mod $m$ Is there any property for: $\frac{x}{y}$ mod $m$ like $\frac{x \mod m}{y \mod m}$ mod $m$ . I hope this fails. I want to ...
3
votes
0answers
91 views

Question about application of Erdős-Kac theorem

My question is whether (*) below can be shown using the Erdős-Kac theorem? I don't think the distinction between $\Omega$ and $\omega$ is important here. For lack of better notation let $\lambda_{r,s}...
2
votes
0answers
52 views

How to prove the $q$-series identity?

How to prove the following identity: $$\sum_{n\ge0}\frac{2q^{n^{2}+n}}{(q)_{n}^{2}(1+q^{n})}=\sum_{n\ge0}\frac{q^{n^{2}+n}}{(q)_{n}^{2}(1-q^{2n+2})}$$
0
votes
2answers
43 views

Is there a simpler way to rewrite this binomial chain?

Consider some binomial chain that looks like this: $$\binom{N}{k_1}\binom{N-k_1}{k_2}\binom{N-k_1-k_2}{k_3}\binom{N-k_1-k_2-k_3}{k_4} \cdots \binom{N-k_1-k_2-\cdots -k_{t-1}}{k_t}$$ Where all ...
0
votes
3answers
73 views

Is there any simple algorithm which can tell if a string is a repeat of its substring?

Is there any simple algorithm which can tell if a string is a repeat of its substring? For example, $1212121212$ is a repeat of $12$, $135746135746$ is a repeat of $135746$.
1
vote
1answer
36 views

Square root in $\mathbb{Z}_p{}^*$

Given a prime $q$, and another prime $p$ = 20q + 1, I am able to find generators in $\mathbb{Z}_p$. Does $-1$ have a square root in $\mathbb{Z}_p{}^*$? Thanks!
3
votes
2answers
207 views

Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
1
vote
1answer
86 views

Decide whether the $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational

Working needs to be shown $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ My guess is to multiply by $\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}$ then we have a rational number but is it enough to prove the ...
0
votes
3answers
2k views

How many prime numbers are also triangular numbers?

I've been trying to figure this out and it's been getting on me myself. I know that $3$ is not just a prime number, but also a triangular number. I'll now add a sequence: Prime numbers: $2, 3, 5, 7,...
1
vote
1answer
68 views

Need help on a proof involving the size of prime factors

Prove that the prime factors of $510510^{510510} + 1$ are greater than or equal to 19. Here is my (incomplete) proof that I need help with: 1. The prime factors of 510510 are 2, 3, 5, 7, 11, 13 and ...
1
vote
1answer
49 views

Find the Value of $n$ Where $15756$ is the $nth$ Member of A Set

It's a question from $BNMO$.It still haunts me a lot. I want to find an answer to this question. Any number of the different powers of $5: 1,5,25,125$ etc is added one at a time to generate the ...
1
vote
1answer
206 views

Minimum number of steps to reach a position

I am on a infinite length number line, Currently at position zero. Each step I can only take R units to right and L units to ...
1
vote
1answer
40 views

Does this define a commutative group?

I've had a look at a binary law between analytic germs of functions, but I don't know if it is a group law, or where to find any reference. Let $\mathcal{H}$ be the space of germs of holomorphic ...
1
vote
1answer
48 views

Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
2
votes
1answer
94 views

Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
4
votes
0answers
115 views

Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
1
vote
0answers
37 views

Proper notation for a generalization of the Liouville $\lambda$ function

Can someone suggest good notation for a simple generalization of the Liouville Lambda function, $\lambda(n) = (-1)^{\Omega(n)}?$ I would like to express that $\lambda_{k,r}(n) = 1$ if $\Omega(n)\...
4
votes
1answer
157 views

How find this limits $\lim_{n\to\infty}(\frac{2}{1^4}+1)(\frac{2}{2^4}+1)(\frac{2}{3^4}+1)\cdots(\frac{2}{n^4}+1)$

How to Find this limit $$\lim_{n\to\infty}\left(\dfrac{2}{1^4}+1\right)\left(\dfrac{2}{2^4}+1\right)\left(\dfrac{2}{3^4}+1\right)\cdots\left(\dfrac{2}{n^4}+1\right)$$ see 1 I remeber ...
2
votes
1answer
118 views

$x^{16}-16 \equiv 0 \mod p$ has a solution for each prime

I have to prove that $x^{16}-16\equiv 0 \mod p$ has a solution for every prime $p$. I already know (from a previous work) that $x^8-16\equiv 0 \mod p$ has a solution for every prime. In my opinion, I ...
0
votes
1answer
102 views

Function $f(n) = 2^{\omega(n)}\mu^2(n)$

Let $$f(n) = 2^{\omega(n)}\mu^2(n)$$, where $\omega(n)$ is number of distinct prime divisors of $n$ and $\mu(n)$ is Moebius function. I want to simplify it. As long as $$ \mu^2(n)=\sum_{d^2|n}\mu(...
4
votes
1answer
109 views

sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers. This to me seems very counter intuitive. Since $(a,p)=1$ the ...
7
votes
1answer
231 views

Reference request for unknown mathematical constant

Below is a plot of $$\dfrac{1}{x}\sum_{n=1}^{x}x\ (\mathbb{mod}\ n)-\dfrac{x}{5.6325}$$ where $5.6325$ is very close to whatever the constant actually is. Does anyone know what this constant ...
3
votes
0answers
110 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus $\sqrt{...
1
vote
0answers
77 views

Dirichlet Characters as Eigenvectors

I have a very concrete question. First an example: If one considers the additive group $Z/(n)$ which is cyclic, the corresponding group characters are the rows of the discrete fourier transform matrix ...
2
votes
0answers
52 views

Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
5
votes
3answers
339 views

What is in clear mathematical terms the definition for a sequence of integers, to be called *random*?

Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms. (1) But what is in clear mathematical terms the established and ...
11
votes
2answers
356 views

Proving that $\frac{\pi ^2}{p}\neq \sum_{n=1}^{\infty }\frac{1}{a_{n}^2}$

we have many formula for $\pi ^2$ as follow $$\frac{\pi ^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}....$$ and $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}....$$ and for $\frac{\pi ...
1
vote
1answer
55 views

Möbius function matrix

In http://en.wikipedia.org/wiki/M%C3%B6bius_function#Matrix_inverse, it is stated, that the values of the Möbius function appear in the inverse matrix of some lower triangular matrix. Does anyone know ...