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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1
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0answers
33 views

Trying to understand a strange regularity found for a ratio of repeated products

I was considering an alternate simplification of $\binom {2 n} {n} $ by pairing the components of one of the denominator factorials with the even terms in the numerator and pairing the other ...
5
votes
1answer
118 views

Solve $3^a-5^b=2$ for integers a and b.

So I have got that (a,b)=(1,0),(3,2) are solutions for the eqations, and maybe the only one.
-1
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0answers
26 views

Non-archimedean balls are/aren't abelian groups

I was reading a book and the author says the following, after defining $B_n(a)=\lbrace x\in\mathbb{Q}_p:|x-a|_p\le\tfrac{1}{p^n}\rbrace$ and $S_n(a)=\lbrace ...
1
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0answers
17 views

Number of solutions to $x^n \equiv a \pmod{2^b}$.

I've been trying to prove the following statement: Let $m \in \mathbb{N}$ and let $2^k$ be the highest power of 2 that divides $m$. Further, let $a$ be an odd integer such that $x^m \equiv a ...
4
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3answers
103 views
+100

the first $2k$ terms of the power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
0
votes
1answer
53 views

Number theory practice exam questions

Looking for some help for these two practice problems for my exam. I'll explain to you what I have so far and my ideas. So for this problem, I solved part (a) using induction, it wasnt too tricky. ...
8
votes
2answers
78 views

Reworking $\sum_{n \leq x} \frac{1}{n^s}$, where $n$ is relatively prime to some fixed $k$

For a fixed integer $k \geq 1$ and real $s>0$ I want to rework the partial sums $$\sum_{\substack{ n \leq x \\ \text{gcd}(k,n) = 1 }} \frac{1}{n^s}$$ in such a way that I can find an ...
0
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1answer
30 views

Primitive Pythagorean triple

I have no idea how to start proving it. I would appreciate any help.
0
votes
1answer
31 views

Monochromatic Solutions

I recently came across this paper: http://borisalexeev.com/pdf/foxgraham.pdf "On Minimal Colorings Without Monochromatic Solutions To a Linear Equation" Can someone explain in clearer terms what ...
1
vote
1answer
45 views

Is a torsor over a variety a variety?

Let $X$ be an algebraic variety over some field $k$ of characteristic 0. Let $g : Y \to X$ be a $X$-torsor under some linear algebraic $k$-group $G$. Is $Y$ also an algebraic variety over $k$?
0
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1answer
28 views

$\frac{-1}{2}$ Zero of odd powers sum polynomials?

Consider the polynomial $S_k(x) \in \mathbb{Q}[x]$ such that $S_k(n)=\sum_{i=1}^{n}i^k, \forall n \in \mathbb{N}$. Now if i recall correctly the definition it should be that ...
2
votes
3answers
57 views

How I can calculate $ \lim_{n→∞}\frac{\ln(2^{p_{n+1}}-1)}{\ln(2^{p_{n}}-1)} $?

Let $\left\{\, p_{n}\,\right\}$ be the sequence of consecutive primes. $$\mbox{How I can calculate}\quad \lim_{n\ \to\ \infty}{\ln\left(\, 2^{p_{n + 1}} - 1\,\right)\over ...
1
vote
1answer
19 views

Find the sequence {a_n} so that $a_i + a_j$ has the same amount of dividers as i+j, for every positive integer i and j

Find all rising sequences positive integers $a_1,a_2,...$ for which holds tgat for every two positive integers i and j the number i+j has the same number of dividers as $a_i + a_j$. I got that if i=j ...
1
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1answer
30 views

Fermat's little theorem question

I'm studying Number theory (in my spare time) and I need to prove a lemma in order to prove the exercise. The topic is Fermat's little theorem. Well the lemma goes like this: Let's say we have ...
2
votes
1answer
39 views

Is this version of Lagrange's four-square theorem true?

Lagrange's four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1\times a_1 + a_2\times a_2 + a_3\times a_3 + a_4\times a_4$ ...
2
votes
1answer
60 views

On Properties of Exponentially Prime Numbers

A usual prime number is a number greater than $1$ which is not in the form of multiplication of two numbers greater than $1$. We may consider the following natural generalizations: $p>1$ is $+$ - ...
-2
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0answers
15 views

Simplify the following Boolean expression (a ⋀ b ) ⋁c) ⋀ (a ⋁ b ) ⋀c) [on hold]

Simplify the following Boolean expression (a ⋀ b ) ⋁c) ⋀ (a ⋁ b ) ⋀c)
1
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1answer
24 views

Find the sum of all the multiples of 3 or 5 below 1000 (Break down)

I know that this has been posted before but I can't grasp how it actually works. I'm a 16 year old in the 10th grade and am interested in algorithms. I've looked in multiple places on the web but am ...
3
votes
3answers
89 views

find the last two digits of $2^{250}$.

Suppose we want the last two digits of $3^{250}$, one can use the theorem $a^{\phi(n)}\cong 1(\mod n)$ whenever $(3,n)=1$. But instead, if i have $2^{250}$, how do i solve this problem, because here ...
7
votes
1answer
121 views

can anyone give a proof by definition: $11$ is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
0
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0answers
24 views

“Write about accelerating convergence using Aitken's $\Delta^2$ process”

My try to solve it: I need someone who understands this field of maths well to correct my answer if any mistakes are found
1
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0answers
37 views

Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
0
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0answers
23 views

$Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C)\mid D$

I read today that $Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C\mid D$ but I can't find it again, I also can't find any Diophantine equations with 3 variables that doesn't have solutions ...
2
votes
1answer
41 views

Positiveness is a relative notion. [on hold]

Consider a ring $R=\mathbb{Z}[\sqrt7]=\{ a+b\sqrt7 ~| a,b\in \mathbb{Z}\}$ and let $p_1=\{ a+b\sqrt7 ~|a,b\in\mathbb{Z} ~~~ \text{and} ~~~ a+b\sqrt 7 ~~\text{is a positive real number} \}$ ...
1
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1answer
42 views

Accounts of the proof of Fermat's Last Theorem

I would like to collect a set of references to pieces of Wiles' 1995 proof of Fermat's Last Theorem. Has anyone recompiled the proof into another paper? Are there any books or articles that describe ...
0
votes
1answer
33 views

An effcient method of solving a Diophantine equation with 3 variables $Ax+By+Cz=D$?

I'm trying to make an efficient algorithm to find one of the solutions and how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb ...
0
votes
0answers
18 views

Characteristic function of $k$-free numbers

I wondering about the fact that the $\sum_{m^k \mid n} \mu (m)=0$ if $n$ is not $k$-free (that means there is at least one prime $p$ such that $p^k\mid n$.
1
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1answer
39 views

Definition of unipotent linear algebraic groups over non algebraically closed fields

Suppose we have a field $F$ with $\text{char}\ F=0$ and $F$ is not necessarily algebraically closed. What is the definition of a unipotent linear algebraic group over $F$? I'd really appreciate ...
-1
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1answer
24 views

Finding the last two digits of $3^{2010}$ [closed]

Find the last two digits of $3^{2010}$
13
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0answers
144 views
+50

An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
-4
votes
2answers
27 views

Proving quadratic inequalities. [duplicate]

I am trying to prove that $$e^{k+1} ≥ 3 + 3k + k^2$$ with, $$k>2$$ WhatI have done so far: What we are trying to prove is that $$e^n≥1+n+n^2$$ is a true statement. Since $n=3$ holds, this is ...
1
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1answer
32 views

Find all Combinations of 1 and 2 which sums up to k.

I have two numbers $1$ and $2$. I have to print all ordered combinations which sums up to $k$. For example: $k=1$ Its only $1$. $k=2$ It's ${1,1},{2}$. $k=3$ Its ${1,1,1},{1,2},{2,1}$ What ...
1
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0answers
13 views

what is the significance of expressing Theta series as Eta-products?

I'm collection information for my upcoming project regarding eta-products and theta-series. I have the Günter Köhler's book, ...
1
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1answer
55 views

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ $\forall n \in \mathbb{N}$. [duplicate]

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ for every $n \in \mathbb{N}$. I'm trying to use induction on this one, but I'm not sure how to. The base case is clearly true. But when I add ...
6
votes
1answer
60 views
+50

Number of distinct angles that can be formed on a square grid

Given a set of grid points arranged in an $n$ by $n$ square (in 2 dimensions): How many distinct proper (acute or obtuse) angles can be formed having a vertex on one grid point and line segments ...
1
vote
1answer
23 views

possible meaning of a constant

I am sorry if the question is ambiguous. Is there any result closely related to the constant $\frac{1+\sqrt{2}}{2}$? It does not count if you say " it is a root of $x^2-x-1/4=0$".
2
votes
1answer
23 views

Sets being subsets of each other

Can a set A be a subset of a set B and B be a subset of A? I know they would just be equal, but are you able to classify them as being subsets of each other?
2
votes
3answers
43 views

Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)?

In the expression $\frac{1}{b}\pmod m$, where $(b,m)=1$, is $\frac{1}{b}$: a) a rational number (and so rational numbers are defined in modulo arithmetic using multiplicative inverses)? b) just ...
0
votes
1answer
25 views

Are the sets $\{\text{fractional part of } n\sqrt{2} : n \in \mathbb{N}\}$ and $\{(m/n)\sqrt{2} : m,n \in \mathbb{N^+}, (m/n)/\sqrt{2} < 1\}$ equal?

If so, what would be a good strategy for showing that this is true? The first set is the set of the fractional parts of integer multiples of $\sqrt{2}$. The second set is the set of all rational ...
1
vote
2answers
20 views

Finding a polynomial mod 5: What did they do here?

I'm reading through this solution for the problem below. The given solution is this: What I don't quite understand is how they got say from $\frac{1}{3}(X-2)(X-4)$ to be 'equal' to ...
3
votes
1answer
31 views

Finding a lower bound to the probability that a number will be shown to be composite?

Given the following method to decide whether a number $m$ is prime or not: Choose a random number $1<a<m-1$, and check whether $a^{m-1} = 1 \mod m$. If its equal, return true, otherwise - ...
1
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0answers
23 views

Prove a property of the divisor function (Part 2)

Further to this MSE question, I would like to pose a follow-up inquiry: If $n \in \mathbb{N}$ and $(\sigma(n) - n) \mid (n - 1)$, does it follow that $n$ and $\sigma(n)$ would have to be coprime, so ...
1
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1answer
32 views

What are the orders of modulo 9 and modulo 28? [closed]

I know what is the order, but unfortunately I can't calculate it. Please help me if you can, thank you very much.
2
votes
0answers
27 views

A number theoretical problem

How many pairs $(A,B)$ are there up to $n$ such that $\gcd(A,B)=B$ and $B^2\neq A$? If we consider $n=5$, we have $25$ possible pairs, $(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(1,5)$, $(2,1)$, ...
1
vote
1answer
53 views

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks
0
votes
1answer
32 views

Base 8 and Base 9 Representations [closed]

A positive integer N greater than 1 is described as special if in its base-8 and base-9 representations, both the leading and ending digit of N are equal to 1. What is the smallest special integer in ...
1
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2answers
22 views

Which property of Jacobi Symbol is used $\left(\frac{x^e}p\right)\left(\frac{x^e}q\right)=\left(\frac xp\right)\left(\frac xq\right)$

While deriving a Jacobi symbol, this derivation is done in a document, but did not explain the property that is used. $p,q$ are prime, $e$ is odd.
1
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0answers
24 views

On counting number pairs having a specific greatest common divisor.

I wanted to count natural numbers $k$ not exceeding the fixed $n \in \mathbb{N}$ and having a greatest common divisor $\gcd(n,k) = d$ naturally for some $d \mid n$. In more mathematical terms: $$ ...
0
votes
1answer
80 views

Numerical value of $\sum_{p \in \mathcal P} \frac1{p\ln p}$

In this question we determine that the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges, where the sum runs over primes. As I see the convergence is really slow. The partial sums for given ...
4
votes
2answers
155 views
+50

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...