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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231 views

How many ways are there of coloring the vertices of a regular $n$-gon

How many ways are there of coloring the vertices of a regular $n$-gon with all $p$ colors ($n,p \ge 2$), such that each vertex is given one color, and every color isn't used for two adjacent vertices? ...
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1answer
185 views

Matrices that satisfy $AB = 0$ and $A^2 = B^2$

I want to make a set of matrices that satisfies all the following: 1) $A^2 = B^2$, $C^2 =D^2$..... where $A,B,C,D...$ are matrices 2) $AB = 0$, $CD = 0$..... 3) All matrices in the set commute. 4) ...
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1answer
371 views

System of 3 linear congruences

Find all solutions: $$\begin{cases} x\equiv 39 \mod(189) \\ x\equiv 25 \mod(539) \\ x\equiv 399 \mod(1089) \end{cases}$$ But $189=3^3\cdot7$, $539=11\cdot7^2$ and $1089=3^2\cdot 11^2$, so I can't ...
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2answers
268 views

Function with a Modular Inverse

For a combinatorics problem I have a function, $h(x)$ that is always divisible by five, but it is calculated in pieces, e.g. $h(1) = 43 + 7$. The final function that I need is $f(x) = (h(x) / 5) ...
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3answers
146 views

Properties of Mediants

If $\frac{a}{c} > \frac{b}{d}$, then the mediant of these two fractions is defined as $\frac{a+b}{c+d}$ and can be shown to lie striclty between the two fractions. My question is can you prove ...
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59 views

series: can the result be zero for a continuous interval of its argument?

I'm considering the series $$ f_c(x) = \sum_{k=c}^\infty \left( c^{k-1} \binom{k}{c} \cdot \prod_{j=1}^{k-1} (x-1/j) \right) $$ where the parameter $c \in \mathbb N ,c \gt 0$ and fixed for a certain ...
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1answer
229 views

Unique Decomposition of Primes in Sums Of Higher Powers than $2$

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma. What is known about sums of $n$ higher powers resulting in ...
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2answers
162 views

Proof that a sum of the first period of powers of integer roots of rationals is irrational

I'm looking for a simple proof or a reference to any proof that For $j \in ℤ$, $0<j<m$, when each $k^j \notin ℚ$ and each $d_j \in ℚ$, $d_j \neq 0$, then $\sum_{j=1}^{m-1} d_j k^j \notin ℚ$ ...
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81 views

Binary forms of degree n

Im trying to show that the binary form $x^{n−1}+x^{n−2}yα+x^{n−3}y^2α^2+...+y^{n−1}α^{n−1}$ is bounded below by $ c*y^{n−1}$ where c is some explicit constant. For the case n=3 this is fairly easy, ...
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527 views

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
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1answer
124 views

Producing infinite family of transcendental numbers

Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically ...
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2answers
392 views

twin prime conjecture

Whether I am correct or wrong I don't know. If there are any corrections, please let me know. Let $p_n$ = product of all primes. (of course we can go still beyond as we know $p_n$ is infinite). Now ...
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1answer
251 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 ...
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2answers
312 views

If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$?

How to prove that: $\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number For example: $\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$ I know ...
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247 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
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3answers
370 views

Patterns in Sequences

I've heard in a movie that for any sequence of numbers, there is a nice formula for generating that sequence. So, for example if I write: 1,2,1,2,3,3,1,2,3,1,2,4,... There is a formula for ...
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362 views

Diophantine equations: ternary forms

Thue proved that all Diophantine equations consisting of an irreducible binary form (cubic or higher) equal to a constant, i.e., $$c_nx^n+c_{n-1}x^{n-1}y+\cdots+c_oy^n=k$$ ($n,k$ fixed) have finitely ...
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117 views

Two succeeding integers in $\left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$ for an odd n, and the Jacobi symbol of the latter one

Given an odd integer $n$, I want to find out if there exists two succeeding integers, $1\leq m-1<m\leq n-1$ s.t both are invertible (i.e $m,m-1\in \left(\mathbb{Z}/n\mathbb{Z}\right)^{*}$) and also ...
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1answer
207 views

Prime numbers which solve $2^s=1\pmod p$

Here we define those primes $p$ for which $\operatorname{ord}_p(2)=s$, where $s$ is the minimum of the set $S$ of all divisors $d\mid p-1$ such that $2^d-1\geq p$. For example: for $p=7$, $s=3$, ...
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211 views

$(a^n +b^n)/((ab)^{n-1}+1)$ is a perfect $n^{th}$ power

Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer? Another question: ...
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110 views

Is that series-transformation known in the context of divergent summation

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
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237 views

How do I compute x^(-2) mod m?

I've written Haskell function, that helps me to compute $x^{-1} \mod m$. ...
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320 views

Finding the Value of $\sum\limits_{n=1}^{p-1} [\sqrt{np} \ ]$

How does one find the value of the sum : $$\sum\limits_{n=1}^{p-1} [\sqrt{np} \ ]$$ where $p$ is a prime such that $p \equiv 1 (\text{mod} \ 4)$. If i remember correctly, i got this sometime back, ...
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206 views

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} - y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out and this came out to be true from $n=1$ to ...
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29 views

Natual density inside a subsequence

Let $S \subset \mathbb N$ be a subset. The natural density is defined as $$D(S) = \lim_{n \to \infty} \frac{|E \cap \{1, \cdots, n\}|}{n}$$ whenever this limit exists. So question is the ...
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30 views

How to find the sum of digits of $p^q$ when $p$ and $q$ are large integers?

Is there any formula to get direct value for this function. $F(p,q)$ = sum of digits in $p^q$ I know that i can compute $p^q$ and sum up the digits. But I want to find it when $p$ and $q$ are big ...
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20 views

Any results to generalize weighted sieve to three parameters?

In Chen's theorem on Goldbach conjecture , he used two parameter weighted sieve method, and he proved every even number can be represented as a prime number and an almost prime ( 1 + 2 ). Are there ...
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82 views

Proof of simple relation involving near primes?

Motivation (can skip!). (*) $\sum\log n \approx n\log n-n,$ and $$\sum\log n = \sum_{p_1\leq n} \log p_1+\sum_{p_2\leq n} \log p_2+...+\sum_{p_m\leq n} \log p_m$$ in which $p_k$ are numbers comprised ...
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41 views

An LCM related problem

If $A=\{27,42,30,94\}$ is a set, then all possible subsets from the set will be $\{27\}$, $\{42\}$, $\{30\}$, $\{94\}$, $\{27,42\}$, $\{27,30\}$, $\{27,94\}$, $\{42,30\}$, $\{42,94\}$, $\{30,94\}$, ...
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60 views

Number of roots of $x^3 \equiv 1 \pmod p$

How to find the number of roots of $x^3 \equiv 1 \pmod p$ in an interval $[a,b]$? For instance, let $p=31$ and $[a,b] = [1,100]$ so the equation becomes $x^3 \equiv 1 \pmod {31}$
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63 views

Prime Reflections

How would you describe the following pattern?: For each primorial from 30 onward, there exists a pattern in the arrangement of the prime factors of the composite numbers which I call "the mirror ...
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45 views

Exact Equivalence of Legendre's Conjecture Impossible?

If the upper bound for the prime gap above $n$ is such that $n$+4$\sqrt{n-1}$$\geq p$, where $n$ is any given natural number and $p$ is the next prime after $n$, then Legendre's conjecture is true. If ...
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70 views

How does $\sum\limits_{i=1}^n i^c \lfloor{n/i}\rfloor$ converge?

How to converge such series : $$y = \sum\limits_{i=1}^n i^c \lfloor{n/i}\rfloor$$ where c can be any constant value, and particularly $i^c = f(i).~$ Also, $f(x)$ is not a Euler's totient function.
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56 views

With odd $\ n>9, \sigma(n) < {11\over 16} e^{\gamma} n \log \log n $?

If that's not the case, do we know anyway some upper bound better than that given by Robin's inequality, since it has been shown that it holds for all odd numbers > 9 ( Choie, YoungJu, et al. "On ...
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224 views

How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$ Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ ...
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1answer
50 views

A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases

As the ring of Gaussian integers is a UFD, this means that $a^2 \mid b^2$ leads to $a\mid b$. Is there any subring of the ring of Gaussian integers with infinitely many elements such that ...
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61 views

modules with several powers $x^{y^z}$

I have an assignment(which gives no credit) which I cant solve so I would like to get some help $3^{{1001}^{1002}}\pmod {43}$ (All original numbers were replaced) Our hint was to use the ...
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33 views

Is ring of Gaussian rationals in unique factorization domain?

Instead of Gaussian integers, let us think about Gaussian rationals, where $a$ and $b$ in $a+bi$ are rational numbers. Then would ring of Gaussian rationals be in unique factorization domain?
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167 views

Is there an odd integer other than 1 that is the sum of its divisors? [closed]

Is there an odd integer other than 1 that is the sum of its divisors (e.g., 6=1+2+3 and 1,2,3 are the divisors of 6)?
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579 views

Count arrangment such that each person wear different tshirt

Few friends are going to a party. Each person has his own collection of T-Shirts. There are 100 different kind of T-Shirts. Each T-Shirt has a unique id between 1 and 100. No person has two T-Shirts ...
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75 views

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$.

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$. My try is let $a$ be a solution of $x^2 \equiv -3 \mod p$. so $a^{p-1} \equiv 1\mod p$. This ...
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128 views

How to find the roots of polynomials in $\Bbb Z_p$

Consider the polynomial $x^{p-1} - 1$ in $\mathbb{Z}_p$. What are its roots / how does it factor? Does this factorization tell you anything about $(p-1)!$ modulo $p$? I'm really stuck on this ...
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197 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
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185 views

Find sum of all permutations

We call two arrays A and B with length n almost equal if for every i (1 <= i <= n) ...
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56 views

The sum of the cubes and the amount of combinations.

Quite simply turned out to solve this Diophantine equation, when he made the assumption that the solutions of these equations symmetric. So given this equation: ...
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82 views

How find this sum $S=\sum_{i=1}^{m}(-1)^{a_{i}}\cdot 2^{m-i}$

let $n$ is give odd positive integer numbers, and postive integer $m$ such $$2^m\equiv 1\pmod n ,2^i\equiv a_{i}\pmod n,0\le a_{i}\le n-1,a_{i}\neq 1,i=1,2,\cdots,m-1$$ Find the sum ...
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4answers
148 views

Show that abc=[ab,bc,ca]*(a,b,c)=(ab,bc,ca)*[a,b,c]

Let a,b $\epsilon$ N. Show that abc=lcm[ab,bc,ca]*gcd(a,b,c)=gcd(ab,bc,ca)*lcm[a,b,c]. How would I proof this?
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53 views

System of quadratic diophantine equations 2

I am looking for a way to simultaneously transform the following four expressions into perfect squares, $1+x_1^2, 1+x_2^2, 1+x_3^2, x_1^2+x_2^2+x_3^2$, i.e. I want to find a rational parametrization ...
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1answer
81 views

help ! discriminant of a polynomial

Let $f(x) = x^3 + ax + b \in Q[x]$ be irreducible. Show that if $\alpha$ is a root of $f$ then $y = \alpha^2$ satisfies $y(y + a)^2 = b^2$ Deduce that $\Delta^2(f) = −(4a^3 + 27b^2)$. I have found the ...
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1answer
56 views

Goldbach's Weak Conjecture

I have a few questions on GWC, as the Wikipedia's page on it appears to be somewhat incomplete. Which of the following two statements is considered as the actual GWC? Every odd number greater than ...