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

learn more… | top users | synonyms (1)

3
votes
1answer
30 views

Asymptotic formula for sums of powers of reciprocals of primes

Is there an explicit asymptotic formula, in terms of $\alpha$, for the expression $$\displaystyle \sum_{p \leq x} \frac{1}{p^\alpha}$$ for $0 < \alpha < 1$? The case $\alpha = 1$ is supplied by ...
40
votes
1answer
794 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
1
vote
1answer
45 views

What books do you recommend on mathematics behind cryptography?

I am currently reading the Book Understanding Cryptography from Cristof Paar. I am enjoying the book but i don't like to scratch the surface when it comes to cryptography. I would like do dig a little ...
3
votes
1answer
31 views

Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
6
votes
4answers
162 views

Prove that $\log_{36} 30 $ is irrational number.

Prove that $\log_{36} 30 $ is irrational number. We can suppose that $\log_{36} 30 $ is rational number. So we have that $\log_{36} 30 = \frac{p}{q}$ where $\gcd(p,q) = 1$. By definition of logarithm ...
3
votes
1answer
267 views

If two primes differ by $n$, then infinitely many primes differ by $n$

A proof I'm writing rests on something I can't prove, probably beyond my knowledge, but it seems right: For any two primes $p_k, p_l$ (not necessarily consecutive) such that the distance between ...
2
votes
0answers
25 views

Nebentypus of contragredient representation

Let $k$ be a local non-archimedean field with ring of integers $\cal O$ and maximal ideal $\frak p$. Let $\pi$ be an irreducible admissible $\infty$-dimensional representation of $\text{GL}_2(k)$ ...
5
votes
3answers
146 views

Diophantine equations and Hilbert's 10th Problem, how did MRDP do it?

I'm having a bit of trouble understanding the Wiki explanation of MRDP's (Matiyasevich, Robinson, Davis, Putnam)'s Theorem, which explains that Hilbert's 10th problem is unsolvable. The MRDP ...
0
votes
0answers
15 views

How to expand powers of multiple pairwise commuting elements in a group [closed]

Let (G, $\ast$) be a group and let n $\in\aleph$. Prove that if $g_1,...,g_k\in G, k\in\aleph$ are pairwise commuting elements of G, then $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
23
votes
2answers
705 views

How to define addition through multiplication?

One might define multiplication $\bullet$ on $\mathbb Z$ as follows: $\bullet: \mathbb Z\times \mathbb N\ni (a,b) \mapsto a+\cdots+a\in \mathbb Z$ where we add $b$ times. But suppose we are in a ...
35
votes
1answer
1k views

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
1
vote
1answer
28 views

How to solve second order congruence equation if modulo is not a prime number

the equation is $x^2 = 57 \pmod{64}$ I know how to solve equations like (*) $ax^2 +bx +c = 0 \pmod{p}$, where $p$ is prime and i know all the definitions for like Legendre's Symbol and all of the ...
52
votes
0answers
767 views

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
1
vote
1answer
61 views

Induction proof involving Euler Totient Function

Let $\varphi$ be the Euler totient function Qi) show that if $r$ is a power of a prime number then $\sum_{d|r} \varphi(d) = r.$ Qii) Show that if $n \geq 2$ then there is a decomposition of n as a ...
4
votes
1answer
88 views

Trying to understand a step in Ramanujan's Proof of Bertrand's Postulate regarding the gamma function

My question relates to this step in the proof here: But it is easy to see that $$\log \Gamma(x)-2\log\Gamma(\frac12x+\frac12) \le \log\left\lfloor ...
1
vote
0answers
17 views

Understanding how to estimate $\pi(x)$ based on Paul Erdos's proof of Bertrand's Postulate

I am reading the 4th Edition of Proofs from the Book. I am not clear on how the proof behind Bertrand's postulate leads to the following statement on page 10 (of my edition): From (2) one can ...
2
votes
0answers
56 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
2
votes
0answers
57 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
25 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 ...
14
votes
2answers
312 views

Numbers that are divisible by the number of primes smaller than them

Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ...
1
vote
0answers
34 views

Clever use of Pell's equation

Find infinitely many triples $(a,b,c)$ of positive integers such that $a,b,c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$are perfect squares. The solution is: Consider the ...
1
vote
0answers
30 views

Strict total ordering

I'm not able to understand how the below relation is example of "strict total order". Consider a set $X = 2^Y$ where $Y = \{1,2,3,4,5,6,7,8,9\}$. The expected order of $X$ is for all $x, y$ ...
4
votes
1answer
43 views

Two question about ${\mathbb Q}(\alpha)$ for $\alpha$ := $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$

Let $\alpha$ = $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$ with $n$ not divisible by $3$. Prove that $[{\mathbb Q}(\alpha) : {\mathbb Q}] = n(n + 3)$. Conclude that $\alpha$ is constructible if and only if $n = ...
0
votes
1answer
12 views

Partition of fractional parts where each sum of them has to be at least 1

Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace ...
6
votes
1answer
178 views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
3
votes
1answer
340 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental ...
2
votes
1answer
62 views

Multiples of 3 and 5. [closed]

If we have the Tartaglia(Pascal) triangle in every row which numers are multiples of 3 which are even and which are multiples of 5?
4
votes
1answer
57 views

Finitely Many Extensions of Fixed Degree of a Local Field

How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the ...
8
votes
1answer
220 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
16
votes
2answers
224 views

Möbius function of consecutive numbers

This question arose from a problem in Niven & Zuckerman's book "Introduction to the Theory of Numbers". In the chapter that the authors introduce the Möbius function, the first exercise is the ...
1
vote
1answer
22 views

Understanding Bézout's identity

I'm trying to understand a proof of Bézout's identity ($gcd(a,b)=$ smallest linear combination of $a$ and $b$), and I'm having some trouble following the last step. The proof goes by: Let $m=sa+tb$ ...
1
vote
2answers
40 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
4
votes
1answer
40 views

Integer solutions to a two variable equation.

For $m, n \in \mathbb{Z}$, show the only integer solutions to $f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}}$ are $f(1, 2) = -7$, $f(0, 1) = -1$, and $f(0, 2) = 1$. More ...
5
votes
1answer
76 views

Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?

Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$. Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate for $\pi(x)$ can ...
5
votes
3answers
101 views

Finding the possible Least Common Multiples of of numbers with Highest Common Factor 8

The Highest Common Factor of two numbers is 8. Which one of the following can never be their Least Common Multiple? The choices are as follow: A. 8 B. 12 C. 60 D. 72 The answer key states ...
5
votes
0answers
63 views

Which prime gaps are known to exist [duplicate]

It is easily proved that prime gaps can be arbitrarily large by constructing the sequence of composites $(n+1)! + 2, (n+1)! + 3, \dots, (n+1)! + (n+1)$, which are divisible by $2, \dots, n+1$ ...
0
votes
1answer
75 views

Integers Positioned Around a Circle [duplicate]

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of n and the product of any two adjacent numbers in the ...
3
votes
1answer
183 views

Chebyshev function identity

given the Chebyshev function $$ \sum_{n \le x} \Lambda (n) = \Psi (x) $$ with $$ \Lambda (n) = \log p $$ for $ n=p^{k} $ and $ 0 $ otherwise is then true that (i think i saw it in apostol book) ...
0
votes
2answers
30 views

all the squares in the multiplicative group $\mathbb{Z}_n^*$. [closed]

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
5
votes
2answers
57 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
-1
votes
1answer
53 views

On no. of solutions of product of positive integers equal to sum [on hold]

$n \ge 2$ be an integer , let $a(n)$ be the no. of solutions in positive integers of $x_1+x_2+...+x_n=x_1x_2...x_n ; x_1 \le x_2 \le ... \le x_n$ , then is it true that $a(n+1)=1 \implies n$ is ...
0
votes
0answers
120 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
11
votes
2answers
382 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
1
vote
0answers
28 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
3
votes
1answer
43 views

Definition of field of fractions of $p$-adic integers

I am given this definition of the field of fractions of the p-adic integers: $$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$ How can I show that: $Q_p$ consists of the sums of the ...
1
vote
1answer
76 views

How can we show that it is an integer 5-adic number?

Show that the number $\frac{3}{8}$ is an integer $5$-adic and calculate the first five positions of its power series in $\mathbb{Q}_5$. Could you explain me how we can conclude that $\frac{3}{8}$ is ...
0
votes
0answers
35 views

Identity of the $p-$norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
0
votes
0answers
35 views

How do we solve this congruence?

I am looking at the proof of Hensel's Lemma. Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots+ a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a p-adic $\alpha_1 \in \mathbb{Z}$ such that: ...
1
vote
1answer
27 views

Chevalley's theorem proof

I'm trying to prove Chevalley's theorem stating that $$ \text{If } f \in \mathbb{Z}[x_1, \dots, x_n] \text{ is a form of degree } r < n \text{,}$$ $$ \text{then there exists a nonzero solution of ...
2
votes
0answers
64 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...