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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1answer
71 views

Looking for references on results on powers of primes dividing $y^n-1$

For a prime $p$ and positive integer $n$, let $E(n,p)$ be the greatest $k$ such that $p^k \mid n$, and $E(n,p) = 0$ if $p \nmid n$. Let $E(n) = E(n, 2)$. A number of years back, I proved the ...
7
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1answer
439 views

Proof: If $n \in \mathbb{N}$ is not the sum of two squares, then $n$ is also not the sum of two rational squares

I have to prove the following: If $n \in \mathbb{N}$ is not representable by the sum of two squares, then $n$ is also not representable by the sum of two rational squares. How do I start here? Any ...
8
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1answer
177 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
0
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2answers
192 views

Heuristic Proof of Hardy-Littlewood Conjecture for 3-term Arithmetic Progressions

The Hardy-Littlewood Conjecture for 3-term arithmetic progressions is that $$ \# \{ x,d \in \{1,\ldots,N\} \, | \, x,x+d,x+2d \text{ are all prime} \} \sim \frac{3}{2} \prod_{p > 2} ...
2
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2answers
246 views

Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as $$ c_p = \begin{cases} 2 & \text{ ...
3
votes
1answer
182 views

surjectivity of group homomorphisms

I don't know if the next thing is true, but I'm not able to find a counterexample: suppose you have a surjective group homomorphism of finite groups $f:G \rightarrow G'$ and normal subgroups $H ...
2
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1answer
193 views

How to go from Fermat’s little theorem to Euler’s theorem thought Ivory’s demonstration?

Ivory’s demonstration of Fermat’s theorem exploit the fact that given a prime $p$, all the numbers from $1$ to $p-1$ are relatively prime to $p$ (obvious since $p$ is prime). Ivory multiply them by x ...
6
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1answer
270 views

Does the inverse of this matrix of size $n \times n$ approach the zero-matrix in the limit as $\small n \to \infty$?

Fiddling with another (older) question here I constructed an example-matrix of the type $\small M_n: m_{n:r,c} = {1 \over (1+r)^c } \quad \text{ for } r,c=0 \ldots n-1 $ . I considered the inverse ...
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1answer
114 views

Quadratic form over the dyadic numbers

I would like to know whether $q=\langle 3,3,11\rangle$ (a diagonal ternary form) represents $2$ over $\mathbb{Q}_2$ (i.e. whether there exist $x,y,z\in\mathbb{Q}_2^\times$ such that $q(x,y,z)=2$). I ...
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1answer
191 views

On the asymptotic behavior of a function related to the number of distinct prime divisors

Let $\omega(n)$ be the number of distinct primes dividing $n$. For $x\in(0,1)$, let $\varphi(x,n)$ be the number of positive integers $m\leq xn$ which are prime to $n$. Show that ...
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1answer
69 views

Continued Fraction: Why do we get with $\gamma \in \mathbb{R}\setminus \mathbb{Q}$ the CF $\frac{1}{\gamma}=\langle0;a_0,a_1,\dotsc\rangle$

I have a question concerning continued fractions: If we have $\gamma \in \mathbb{R} \setminus \mathbb{Q}$ and $\gamma=\langle a_0;a_1,a_2,\dotsc\rangle$. Why do we get $$\frac1\gamma = \langle ...
1
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2answers
111 views

Every number $n^k$ can be written as a sum of $n$ distinct odd numbers

I wish to prove that for $n,k\in\mathbb{N} > 1$, we can always write $n^k$ as a sum of $n$ odd positive integers. I have an idea of how to approach this, but my method seems to cumbersome. I am ...
1
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1answer
161 views

A Rough Estimation for the number of square free integers

Show by a sieve argument that the number of square free integers not exceeding $x$ is less than $$x\prod_p\left(1-\frac{1}{p^2}\right)+o(x),$$where the product extends over all primes. I happened ...
0
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1answer
289 views

Primitive Dirichlet Character

Let $\chi$ be the trivial Dirichlet character mod $N$. What is the primitive Dirichlet character associated to $\chi$? Is it just the character on $\mathbb{Z}$ that sends all integers to 1?
8
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1answer
1k views

Is there a way to find the first digits of a number?

Is there a way to find the first digits of a number? For example, the largest known prime is $2^{43,112,609}-1$, and I did sometime before a induction to find the first digit of a prime like that. ...
3
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0answers
144 views

Champernowne-like squares, are there any?

I read about the Champernowne constant on Wikipedia a couple of days ago, and I got curious about something similar: is there some "Champernowne-like" number; that is, a concatenation of all numbers ...
0
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1answer
390 views

An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”

This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$ which are generally called as Pell-conics, so all in this question $K$ refers to ...
7
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2answers
182 views

On a double sum involving prime numbers

$$\sum_{i,j=1}^{\infty}\left[\frac{x}{p_ip_j}\right]=x\sum_{p_ip_j\leq x}\frac{1}{p_ip_j}+O(x),p_i< p_j$$ where $p_i$ is the $i$th prime, and "[ ]" represents the largest integer not ...
18
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3answers
948 views

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational. I can prove that $x$ is irrational by showing that it's a root of a polynomial with integer coefficients ...
1
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1answer
84 views

$\mathbb{Q}(t_1,…,t_n) \cap \overline{\mathbb{Q}}$

Let $\overline{\mathbb{Q}}$ the algebraic closure of $\mathbb{Q}$, and $K$ a field extension of $\mathbb{Q}$ (not necessarily algebraic) such that $[K:\mathbb{Q}]= \infty$. Let $t_1,...,t_n \in K$, ...
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3answers
486 views

Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$

The equation is $(x!)(y!) = x!+y!+z! $ where $x,y,z$ are natural numbers. How to find out them all?
0
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1answer
85 views

Representing complex numbers with nested exponentiation of rationals

Define $L_0=Q$ $L_1=\lbrace x \in C; e^{x} \in L_0 \rbrace$ $L_{-1}=\lbrace x \in C; \ln{x} \in L_0 \rbrace$ $L_{n+1}=\lbrace x \in C; e^{x} \in L_n \rbrace$ $0$ is in $L_1$ and $L_0$. Do any ...
5
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1answer
389 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
4
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1answer
72 views

Generating Functions: how do I get my answers in terms of differential operators?

I'm reading and enjoying "generatingfunctionology". What a great fun book! But, I'm having some difficulty with the exercises. For example, take the series $a_n = n^2$ I'd like to find the Generating ...
7
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3answers
1k views

RSA: How Euler's Theorem is used?

I'm trying to understand the working of RSA algorithm. I am getting confused in the decryption part. I'm assuming $$n = pq$$ $$m = \phi(n) = (p - 1)(q - 1)$$ E is the encryption key $\gcd(n, E) = ...
6
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3answers
167 views

Showing $\pi(ax)/\pi(bx) \sim a/b$ as $x \to \infty$

I'm having a bit of a problem with exercise 4.12 in Apostol's "Introduction to Analytic Number Theory". I don't think it's supposed to be a very hard exercise, it's the first one in its section ...
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1answer
133 views

Right angle triangles with bases $c/2^{2^n}$

Let's have a right angle triangle where $a=5$, $b=4$, $c=3$. Is it possible to create an infinity of right angle triangles with rational sides from the above triplet, with bases equal to $3/2^{2^n}$ ...
9
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2answers
253 views

Erdős: Sum of rational function of positive integers is either rational or transcendental

I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its ...
2
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0answers
102 views

Dedekind Spectra

Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum which is somehow ...
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1answer
273 views

Number theory conjecture [closed]

Let us observe the following pattern $N - p_1 = m_1, N - p_2 = m_2, \ldots , N - p_r = m_r$; take $p_1 = 3$ and $p_2 = 5,\ldots$ notice that $p_r$ is the larger prime less than or equal to square ...
2
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3answers
335 views

How to prove that if $a\equiv b \pmod{2n}$ then $a^2\equiv b^2 \pmod{2^2n}$

What I have done is this: $a\equiv b \pmod{2n}$, $a=b+c\times2n$, for some $c$, $a^2=b^2+2b\times c\times2n+c^2\times2^2n^2$, $a^2-b^2=(b\times c+c^2n)\times4n$, then $a^2\equiv b^2\pmod{2^2n}$. ...
6
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1answer
285 views

Primes that ramify in a field

Consider the number field $L/\mathbb{Q}$. I know that the only primes $p$ that ramify over $L$ are the ones that divide $\Delta_{L}$, the discriminant of $L$. But what if I can't compute $\Delta_{L}$? ...
0
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1answer
71 views

why did they use 9 in step 3 as the 4-th root of unity

Im doing a problem for al algorithms class to multiply 2 polynomials using FFT, and am confused as to why they picked 9 in step 3 in this document: ...
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0answers
157 views

Forcing and divisibility

I'm going to bring together a couple of seemingly unrelated questions that I've asked here. This may be silly. Or maybe not? Imagine that $n$ is some sort of infinitely large integer, and thus so ...
5
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3answers
250 views

Evaluate $d(n!)$

An exercise: Using the prime number theorem find an asymptotic expression for $d(n!)$ where $d$ is the number of divisors.
0
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2answers
215 views

computing primes

As per my knowledge, I have seen the only following functions which will produce primes for $n$: $n^2 - n + 41$ $n^2 + n + 41$ Of course both functions faile for $n = 41$ due to the polynomial ...
13
votes
5answers
3k views

What is the fastest growing total computable function you can describe in a few lines?

What is the fastest growing total computable function you can describe in a few lines? Well, not necessarily the fastest - I just would like to know how far an ingenious mathematician can go using ...
11
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2answers
339 views

How rare are the primes $p$ such that $p$ divides the sum of all primes less than $p$?

This is just for fun! The title pretty much says it all. It's probably a very difficult question. Up to the $40,000^{th}$ prime $(479909)$, I have found only $5$, $71$ and $369119$ with this ...
2
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0answers
105 views

Does there exist a number field with the following property

Let $\overline{\mathbf{Q}}\subset \mathbf{C}$ be the field of algebraic numbers. Does there exist a number field $K$ with the following property? There are embeddings $\sigma,\tau:K\to ...
3
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1answer
265 views

Inertia groups generate Galois Group

While reading a paper about the Kronecker-Weber Theorem, I noticed a theorem saying that for a Galois extension $K/\mathbb{Q}$, its Galois group is generated by $I_p$s, being the inertia groups of ...
2
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4answers
254 views

Form of rational solutions to $a^2+b^2=1$?

Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?
0
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2answers
99 views

Questions regarding congruences of multiples and relatively prime numbers

If we say $n=p_1^{\alpha_1}\times p_2^{\alpha2}\times \cdots \times p_k^{\alpha_k}$, where $p_i$ are prime numbers, $\alpha_i$ are natural numbers, can or can we not say that: Choose a $p_i$ such ...
0
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1answer
175 views

Vinogradov's equidistribution theorem

Is it true that $(\alpha p_k)$ is equidistributed on $[0,1)$ mod 1 (Vinogradov) $\Leftrightarrow$ $(p_k)$ is equidistributed on $[0,2\pi) $mod $2\pi$ ? $p_k$ is the kth prime and $\alpha$ is an ...
2
votes
1answer
148 views

Reciprocal of a continued fraction

I have to prove the following: Let $\alpha=[a_0;a_1,a_2,...,a_n]$ and $\alpha>0$, then $\dfrac1{\alpha}=[0;a_0,a_1,...,a_n]$ I started with ...
3
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1answer
129 views

$\pi$ and $e$ as irrational linear combinations

Let $S=\{m\cdot n^r\mid m,n\in\mathbb Z,r\in\mathbb Q\}$ Can $e$ or $\pi$ be written as a finite sum of elements of $S$? Can $\pi=xe$, with $x$ algebraic?
4
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1answer
177 views

2-dimensional $\ell$-adic representations [closed]

In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or ...
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1answer
131 views

proof of theorem

How to prove the following theorem. Could you explain. Given the number $A = \langle a_n, a_{n-1}, \dots, a_0 \rangle_{10}$ and the modulus $m$ such that $(10, m) = 1$ from the sequence ...
0
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1answer
370 views

Resurrection of my Tamagawa numbers Question, to understand the Formulation of BSD

My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not ...
2
votes
3answers
230 views

Kronecker-Weber Theorem

I'm stuck with an article "A simple proof of Kronecker-Weber Theorem" on this website. On page 7, the author proofs that $\mathbb{Q}_p((-p)^{\frac{1}{p-1}}) = \mathbb{Q}_p(\zeta_p)$. While I ...
3
votes
1answer
345 views

Question about a proof in Apostol's “Introduction to Analytic Number Theory”

The question is about the proof of Theorem 2.17 (Page 36) of the book Introduction to Analytic Number Theory by Apostol: Theorem 2.17. Let $f$ be multiplicative. Then $f$ is completely ...