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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Closed Form of n(mod7)

For an integer n,what is the closed form, if it exists, of n(mod7)={0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,...,n(mod7)}? The closed form of n(mod8) uses trigonometric expressions so I would ...
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3answers
40 views

Does Bezout's lemma work both ways.

I know that if $a$ and $b$ have a highest common factor $h$ then you can write $h=ax+by$ for some $x,y \in \mathbb{Z}$ but how about if you can write $h=ax+by$ for some $x,y \in \mathbb{Z}$ then can ...
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1answer
20 views

pth root of unity in $p$-adic field

It is well known that $\mathbb{Q}_p(\mu_n)$ is a totally ramified extension of degree $(p-1)p^n$ if $\mu_n$ is a primitive $p^n$th root of unity. However how true is this statement for a finite ...
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2answers
31 views

Proof with 3 coprime numbers.

Given is $X$ is coprime to $Z$ and $Y$ is coprime to $Z$ prove $XY$ is coprime to $Z$. I know you can use Bezout's lemma to say $1=aX+bZ$ and $1=cY+dZ$ but I don't know how to actually do the proof. ...
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0answers
19 views

Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p-$adic $a_1 \in \mathbb{Z}$ such that $$F(a_1) \equiv 0 \mod p \mathbb{Z}_p ...
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2answers
13 views

Finding $2m+1=2\alpha k+\alpha^2$ quickly

Given some positive integer $m$ I'm looking for all solutions $\alpha,k>0$ to $2m+1=2\alpha k+\alpha^2$ with $0<k^2<2m.$ Right now I'm finding these by looping over each of these possible $k$ ...
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0answers
18 views

There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p+m, m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$. ...
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0answers
11 views

The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$

Proposition Let $P(X,Y) \in \mathbb{Z}[X,Y]$. The following propositions are equivalent: The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$. For each $n \geq 0$ the equation $P(X,Y)$ has a ...
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1answer
62 views

Inverse limit of $\mathbb{Z}/n\mathbb{Z}$

I know that this is well-known fact that $$\lim\limits_\leftarrow\mathbb{Z}/n\mathbb{Z}=\prod\limits_p\mathbb{Z}_p,$$ however I don't know the rigorous proof of this. Can anyone give me the ...
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0answers
25 views

Short intervals with all numbers having the same number of prime factors

How to prove that for some $k, n_0$, for all $n \ge n_0$ it is never the case that all integers in $\{n, n+1, \dots, n + \lfloor (\log{n})^k \rfloor\}$ have exactly the same number of prime factors ...
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0answers
25 views

analysis/number theory study group (online) [on hold]

I plan on studying analysis from landau, rudin probably others and am looking for people (hopefully more than 1) where we could solve theorems/problems and ask each other questions. Online ...
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0answers
19 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a sequence of natural numbers (i.e. this is a probability space of sequence of natural numbers sometimes used in ...
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2answers
52 views

Proving that $\sin1 $(radian) is irrational without using Taylor Series Expansion.

In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational, and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but ...
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3answers
43 views

Counting the number of $\mathbb{F}_q$ points on a homogeneous polynomial

This is an area of number theory that I am not too familiar with and I would appreciate any assistance! Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic not 2 or 3. I have the ...
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0answers
26 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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1answer
64 views

Intuituive reason why Fermats last theorem holds

I am unsure of whether it is normal, but to me, intuitively Fermats last theorem should not hold. If anyone intuitively believed it to be correct, why? Can someone explain so I understand ...
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1answer
23 views

How do we deduce that $\mathbb{Q}_p=\{ p^mu\mid u \in \mathbb{Z}_p^{\star}, m \in \mathbb{Z}\} \cup \{0\}$?

We know that each element $x$ of $\mathbb{Z} \setminus{\{0\}}$ has a unique representation of the form $x=p^m u\mid m \in \mathbb{N}_0, u \in \mathbb{Z}_p^{\star}$. $$\mathbb{Q}_p=\left \{ ...
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2answers
52 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...
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1answer
33 views

What is the probability that a random K-bit odd-number is prime?

Is it $e/K$? In an experiment that created 1000 random RSA-2048 key-pairs, 2000 random 1024-bit primes were created. It turned out that $727,709$ random candidates were generated, to create 2000 ...
2
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1answer
22 views

Suggestion to a book about number theory

What I am looking for is a book that contains "infinitely many problems", starts from the easiest to high level(that can be found in national and even international olympiads). Are there such books, ...
2
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0answers
36 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
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21 views

Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin ...
2
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2answers
29 views

a root of some polynomial over finite field

I think this is a really basic question, but it had been a little while since I dealt with this material and I was hoping to get a bit of assistance here. Let $q = p ^{2M}$ for some prime $p$ and $M ...
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0answers
58 views

Idele for a rational number $q=\frac{63}{550}$ [on hold]

Wikipedia, in its article "p-adic number", has taken an arbitray number $x= \frac{63}{550}$ to show the p-adic absolute value with respect to different primes. Obviously, the p-adic absolute value is ...
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1answer
13 views

solve $k(k-1) \geq \ln2*2m$ for k

My Question is related to the birthday problem. Starting at $e^{-\frac{k(k-1)}{2m}} \leq 0.5$ i used $ln(x)$ on both sides and multiplied by $-2m$ to get $k(k-1) \geq \ln2*2m$ According to my ...
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1answer
52 views

The diophantine equation $a^7+b^7=7^c$

Determine all the triples of positive integers $a,b,c$ such that $a^7+b^7=7^c$.
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1answer
32 views

If $a+bi$ is in $E_k$ then $a-bi$ is also in $E_k$?

I'm currently studying the properties of the Motzkin sets $E_k$, $k\in\mathbb{N}\cup\{0\}$ of the ring $\mathbb{Z}[i]$. The definition of $E_k$ is as follows: $E_0=\{0\}$, $E_1=$units of ...
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2answers
45 views

Understanding why the public exponent $e$ is chosen the way it is in RSA

I am trying to get a better understanding of RSA. At the moment I am unable to understand the difference between the correctly chosen value of the public exponent $e$ and other possibilities ...
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0answers
28 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
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2answers
38 views

Solving Linear Congruences With The Euler Totient Functio

I've been asked to calcualte $15^{123456789012345}$ mod $2500$. Now I worked out $\varphi(2500)=1000$ but I can't use Euler's theorem here because $1000$ and $2500$ aren't coprime. Can anyone offer ...
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1answer
48 views

Every element of field $F_q$ has $k$th root if and only if $\gcd(q-1,k)=1$

Help me please to prove that: For any $k \in \mathbb{N}$ each element of field $F_q$ is the $k$-th power of some element from that field if and only if $GCD(q-1, k)=1$. My approach Let's look ...
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1answer
29 views

Solving Linear Congruences With Euler Totient Function

I've been asked to solve the following congruence $x^{1667}\equiv2$ $mod$ $2500$. Am I right in saying there's no solution modulo 2500 to this congruence since even though 1667 is coprime to ...
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0answers
29 views

Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
3
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1answer
48 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
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0answers
32 views

Number theory and Group theory [on hold]

Can you give me any task which contains Number theory and Group theory?
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2answers
99 views

A number $N$ is a $k$-nacci number if and only if …

For $k\ge 2\in\mathbb N$, one can define the $n$-th $k$-nacci number $f_k(n)\ (n=0,1,\cdots)$ as $$f_k(0)=f_k(1)=\cdots=f_{k}(k-2)=0,\ \ ...
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0answers
27 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
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3answers
91 views

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any ...
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1answer
37 views

All possible number combinations in decimal represenation of irrational numbers?

This question is directly inspired by "Does Pi contain all possible number combinations?". I would like to state firstly for the record that I have no serious number theory education. I think I ...
3
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1answer
28 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 ...
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0answers
28 views

On a congruence for the number of finite topologies

I am making search about "On a congruence for the number of finite topologies". I have found a paper. I guess it is written in Russian. How can I find English version of this paper ? I am also ...
3
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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 ...
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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 ...
11
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1answer
101 views

Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?

I've been looking for perfect squares that can be represented as $\sum\limits_{k=0}^{n}p^k$. Of course, both $n$ and $p$ should be natural numbers larger than $1$. Searching up to $n=100$ and ...
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0answers
15 views

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

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 $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
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1answer
24 views

Proofed: Every number in the sequence of powers of 2 have $phi = 1/2 * 2^x$

I want to know if it's proofed, that every number which is in the number sequence of the powers of $2$ has an $\phi$ of $\frac12x$.
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1answer
58 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 ...
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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
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0answers
24 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
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1answer
74 views

The number of divisors of a number whose sum of divisors is a perfect square

Let $n$ denote a non-prime whose sum of divisors is a perfect square. I have noticed a few surprising facts on the number of divisors of $n$: It is either prime or semi-prime or $27$ in all cases ...