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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68 views

Provide a proof using the rational roots theorem

Prove that if $c$ is a positive rational and $k$ is a positive integer, then $c^{1/k}$ is either an integer or irrational, using the rational roots theorem.
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1answer
190 views

Square root of 5 in modulo prime field

How can we efficiently find square root of 5 in a mod prime field. By quadratic reciprocity we can argue that 5 is a square in modulo p(prime) is p is square modulo 5. But how exactly can we calculate ...
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2answers
94 views

How find this value $m^2-mn-n^2$

let $$1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots\dfrac{1}{1}}}}}=\dfrac{m}{n}$$ where $m,n$ are positive integer numbers,and such $gcd(m,n)=1$.,and article $1998$ the fractional ...
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2answers
53 views

Prove $\frac{3^{2m+1}+1}{4}$ and $3^{2m+1}-1$ are coprime

For a positive integer $m$, let $n=2m+1$. Then how can I prove $\gcd{(\frac{3^n+1}{4}, 3^n-1)}=1$? Thanks in advance.
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2answers
75 views

how to find the near number to make zero remainder for a quotient?

There is an expression A*x/B, where A and B are given constant (positive integer), x is also positive integer and initially given but I am trying to find the next integer such that A*x is multiple ...
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1answer
105 views

Show that if $(a, b) = 1$, $a|c$ and $b|c$, then $(a · b)|c$. [duplicate]

"Show that if $\;(a, b) = 1\;$, $\;a|c\;$ and $\;b|c$, then $(a · b)|c$." $$$$Show: We know that $$x\mid w \;\;\text{and}\;\; y\mid w \Longleftrightarrow \frac{x\cdot y}{(x,y)}\mid w$$So if$$a\mid ...
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2answers
116 views

how many $0$ does $150!$ have when it transform to base $7$?

how many $0$ does $150!$ have when it transform to base $7$? can someone help me please how can I able to solve this problem.thanks for your help.
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1answer
169 views

Infinitely many primes of the type 5 mod 6.

Problem: Prove that there are infinitely many primes of the type 5 mod 6. My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this ...
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3answers
54 views

find $0 < l < 35$ such that $l^5 \equiv 3\pmod {35} $

I have to find some $0 < l < 35$, such that $l^5 \equiv 3\pmod {35} $. I tried to use suggestions from my previous question, So I tried: $l^5 \equiv 3\pmod {35} $ => $35 | l^5 - 3$, I ...
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1answer
49 views

Show that if $b\mid c$ and $b > \gcd(c, d)$, then $b\nmid d$.

I have no clue about how to go about this question. I feel like I need more info, but I don't know, please help.
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3answers
106 views

Quadratic residues, mod 5, non-residues mod p

1) If $p\equiv 1\pmod 5$, how can I prove/show that 5 is a quadratic residue mod p? 2) If $p\equiv 2\pmod 5$, how can is prove/show that 5 is a nonresidue(quadratic) mod p?
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2answers
565 views

Missing Exercises in Elementary Number Theory by Underwood Dudley.

I'm a beginner in math and I just started studying Elementary Number Theory by Dudley. So far I'm impressed, but I've noticed that the book does not include all the solutions to the exercises they ...
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1answer
51 views

Finding Residue Classes

Can someone explain to me how to find all the elements in $\mathbb{Z}^*_{15}$ Wouldnt the solutions be: $[1], [2], [7], [11], [13]$
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1answer
53 views

Question regarding Legendre symbol and Quadratic reciprocity.

How would determine the value of the following Legendre symbol is $1$ or $-1$? $$\left(\frac{\frac{p - 1}{2}}{p}\right)$$ So far, I've been able to figure out this much: $$\left(\frac{p - ...
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2answers
271 views

Mapping from 1D line to 2D plane: an infinite piece of rope covering 2D plane without self-intersection

I believe I'm looking for a function: $f(x) : \mathbb N \mapsto \mathbb N^2$ and it's inverse $f^{-1}(x) : \mathbb N^2 \mapsto \mathbb N$, a known mapping that can take any positive integer and map ...
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2answers
86 views

Number theory equation(diophantine)!

Find all $x$, $y$ integers s.t. $ 8(x+y)(x^2+y^2)=15(x^2+y^2+xy+1) $ I am new to Diophantine equation, I have tried all kinds of algebraic manipulations but in vain, I have also tried to think of it ...
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2answers
126 views

How to calculate: $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+…+\frac{99}{100} $

How can I calculate value of $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+.....+\frac{97}{98}+\frac{99}{100}$. My try:: We Can write it as $\displaystyle \sum_{r=1}^{100}\frac{2r-1}{2r} = ...
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1answer
64 views

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
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1answer
74 views

Finding $n$ if the given number is perfect square

Find $n$, if $2^{200}-2^{192} \cdot 31+2^n$ is a perfect square. $$2^{200}-2^{192} \cdot 31+2^n = 2^{192}(2^8-31)+2^n = 2^{192}(256-31)+2^n = 2^{192} \cdot 225+2^n$$ For some $m \in \mathbb{N}$, ...
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2answers
691 views

Obtaining wrong quotient when dividing by negative number

I'm having a big moment of ignorance, since my math teacher in college showed us that no one from my class knew to divide two integers right, without knowing anything about the knowledge of my class, ...
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1answer
127 views

Solve another Diophantine equation with 2 variables and odd degree 5

See also the already solved question: Solve a Diophantine equation with 2 variables and odd degree 5 Prove that there are no non trivial integer solutions to the equation $a^{5} -1 = 2b^{5}$
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2answers
232 views

Infinite sum of fractions

Lets formulate the following series of fractions a_1/b_1 ,a_2=a_1+1/ b_2=a_1+b_1+2, a_3=a_2+1/b_3=a_2+b_2+2 and so on.If a_1=2 and b_1=4 we have the following series of ...
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1answer
66 views

Expressing as a continued fraction

How can I express this as a continued fraction? $\left(\frac{2207 + 987\sqrt{5}}{2}\right)^{1/8}$
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3answers
118 views

Primes + Inetvel + conjecture on primes

a) Can we establish a proof, there exists infinitely many primes of the form $n^2$ + 1. Why is the unit digit of such a prime p always 1 or 7? Is there any reasonable procedure or concept for the fact ...
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2answers
122 views

given average score,find the selected numbers

Suppose average score of 46 scores selected from 1,2,3,4,5 is 1.65.Is there any way to find out which are the scores that are selected.
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1answer
102 views

Let $p \ge 5$ be a prime number. Find the largest length of an arithmetic progression satisfying the following

Let $p \ge 5$ be a prime number. Find the largest length of an arithmetic progression, of positive ratio, of positive integers whose terms do not contain the digit $1$ in their p-adic expansion.
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1answer
668 views

Reed-Solomon Code calculation

I have a Reed-Solomon Code which can correct t=2 errors. The generator polynomial is $p(X) = X^3 + X + 1$ and $p(a) = a^3 + a + 1 = 0$ this means $a^3 = a + 1$ What is the degree of generator ...
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2answers
146 views

if $x^2 \bmod p = q$ and I know $p$ and $q$, how to get $x$?

if $x^2 \bmod p = q$ and I know $p$ and $q$, how to get $x$? I'm aware this has to do with quadratic residues but I do not know how to actually solve it. $p$ is a prime of form $4k+3$
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1answer
81 views

Why does this algorithm work (where does it come from) for finding the period of a decimal expansion?

For the period of something like $1/d$ where $d$ is a positive integer, I saw an algorithm repeatedly doing: $$\begin{align*}r &= 1\\ r &= 10r \bmod d \quad\text{ (until } r = 1) ...
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1answer
210 views

The number of multiples of $10^{44}$ that divide $10^{55}$. [duplicate]

Possible Duplicate: Number of Multiples of $10^{44}$ that divide $ 10^{55}$ I want to find out the number of multiples of $10^{44}$ that divide $10^{55}$ from the following options. $(a)\ ...
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3answers
103 views

Determining when $3 \cdot 5^a \cdot 7$ is abundant.

I would like to determine the values of $a$ for which $3 \cdot 5^a \cdot 7$ is abundant. My work so far: $$\sigma(3 \cdot 5^a \cdot 7) > 2 \cdot 3 \cdot 5^a \cdot 7 = 42 \cdot 5^a ...
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1answer
294 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?
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1answer
394 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 ...
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1answer
206 views

The Galois orbit of an algebraic number

Let $\alpha$ be an algebraic number and let $S$ be the orbit of $\alpha$ under the action of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Do we have that $\# S $ is bounded from above by the ...
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3answers
201 views

Diophantine equation

If one solves the Diophantine equation $cx + by = a$; i.e., $cx = a - by = a \pmod{b}$ formally, then the answer is $x = (a/c) - (b/c)y$, but the integer character and information is lost and not ...
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1answer
270 views

Sum of two squares [duplicate]

Possible Duplicate: Prove that $n$ is a sum of two squares? I was reading this and began wondering if there is a general theorem that a number of a given form is the sum of two squares. I ...
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1answer
188 views

Showing $e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$

Let $$ \theta(x) = \sum\limits_{p \leq x} \log{p} \quad \ ; \ \psi(x)=\sum\limits_{n=1}^{\infty} \theta(x^{1/n})$$ then how does one prove $$e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$$
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1answer
72 views

Why does 0-N invert's the sign

http://stackoverflow.com/questions/4113133/changing-a-positive-value-to-a-negative-one why? is this just a computer science problem , or is this a mathematical property.
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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 ...
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1answer
15 views

Number theory,GCD, coprime integers

I am sorry for the bad title but I really can't think of a better one. So I was learning about the euclidean algorithm and I see a statement that is hard for me to understand. In the book that I was ...
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2answers
51 views

Ramification and Quadratic Reciprocity Law

I have a question regarding the follow problem: Show that the prime number 27644437 splits completely in $L = \mathbb{Q}(\sqrt{55})$. From what I understand. This deals with ...
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2answers
44 views

I want to prove $1^{p-1} + 2^{p-1} + \cdots + (p-1)^{p-1} \equiv -1 \mod p$ [closed]

I want to prove that $$ 1^{p-1} + 2^{p-1} + \cdots + (p-1)^{p-1} \equiv -1 \mod p $$ where $p$ is a prime using elementary ways. How can I prove it ?
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1answer
31 views

General Behavior of Euler Totient Function

If we have two integer M and N such that $$GCD(M,N) = k$$ Then what is $$\phi(MN)$$ There is a famous identity which states: $$GCD(M,N)= 1 \rightarrow \phi (MN) = \phi(M)\phi(N)$$ And now I am ...
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2answers
42 views

Calculate the power series

I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$. In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$? If ...
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2answers
32 views

For any integer a, if $6|(3−a)$, then $3| (a−2)$. [closed]

Prove: For any integer a, if $6|(3−a)$, then $3| (a−2)$. I've been trying to work this problem for a while, but missed a day of class and can't seem to work it out.
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1answer
28 views

Prove that for a sequence of people sets $S_1,…,S_d$, $\Delta_i \not = 0$ for all people

We have $k$ people $p_1,...,p_k$, and $d$ people sets $S_1,...,S_d$, where the sizes of $S_1,...,S_d$ can vary between $1$ and $k$ (so each $S_1,...,S_d$ is a set of some people from ...
0
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1answer
45 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
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2answers
50 views

Beautiful little number theory prob

Solving (a,b) + [a,b] = ab for natural a,b. How many possible a's are there. I only know that (a,b)[a,b] = ab. Tried factoring out (a,b), but can't derive from it.
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2answers
63 views

A curious elementary number theory problem

Find all $n$ satisfies $\forall k ((k,n)=1$ $\Rightarrow k^2 \equiv 1 (\mathrm{mod} n))$. For example when n=8, k=1,3,5,7 satisfies the condition. When n=24,k= 1,5,7,11,13,17,19 satisfies the ...
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2answers
38 views

Proving rationality given

Trying to find a proof that $(x^n -1)^{1/n}$ is rational/irrational given $x$ is rational and $n>3$. I've tried searching online and in libraries. It's hard to find.