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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13 views

Count the strings with n0 K zeroes together

Given a string of length N that is made of only 0 and 1's.But some positions of string are '?'.It means their we can put 0 or 1. Now , the problem is we need to count the number of ways to fill these ...
1
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1answer
31 views

Sum over divisors of sum over coprimes

Set $n \in \mathbb{N}$ , $n>1$ . Consider the function $\phi_{1}$ as $$\phi_{1}(n)= \sum _{r=1 \atop gcd(r,n) =1}^{n} r$$ Prove that $$\sum_{d|n} d \cdot \phi_{1}\Big(\frac{n}{d}\Big) = ...
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1answer
27 views

$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
1
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0answers
25 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
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0answers
11 views

2 players cross out numbers from the line [on hold]

There is a line with even number of numbers written on it. Two players cross out this numbers one by one from left or right. At the end they find the sum of numbers they crossed out. The winner is one ...
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1answer
35 views

Prime number theorem lemma: prove that $\psi(x)\sim\pi(x)\log(x)$

I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One ...
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1answer
23 views

Best self study book with answers to selected questions for analytic number theory?

All: Can anyone recommend Best self study book with answers to selected questions for analytic number theory ? If a book have no answers to questions, but if you know if some professors choose the ...
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2answers
27 views

Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$

I am using the Euclidean Algorithm (EA) for proof. Let $a>b$ and by EA we have $$ \begin{align} a=q_0 b+r_1 & & & \text{where }0\leq r_{1}<b \\ b=q_1 r_1+r_2 & & & ...
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1answer
20 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
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2answers
50 views

How find this minimum of the $q$,if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$

let $p,q$ is postive integer,and such $$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$ Find the minimum of the $q$ maybe can use $$95q>36p$$ and $$37p>96q$$ and then find this minimum of ...
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3answers
31 views

Modular calculus and square

I want to prove that $4m^2+1$ and $4m^2+5m+4$ are coprimes and also $4m^2+1$ and $4k^2+1$ when $k\neq{m}$ and $4m^2+5m+4$ and $4k^2+5k+4$ when $k\neq{m}$. Firstly : Let $d|4m^2+1$ and $d|4m^2+5m+4$ ...
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2answers
34 views

Generating function for number of integer solutions, no computer

How do you solve a Generating function for the number of integer solutions with no computer? Use a generating function to solve the number of integer solutions for $$x_1+x_2+x_3=17$$ Where ...
1
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0answers
18 views

Number of distinguishable arrangements of the word INDOOROOPILLY with three different conditions

I have the following three questions on a past final exam, I wanted to ask if I have done everything correctly. Thank you! How many distinguishable arrangements are there for the letters of the ...
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0answers
9 views

Affine cipher and shift cipher

I have the following question: $$An\;affine\;cipher\;with\;key\;K(0,b)\;is\;equivalent\;to\;a\;shift\;cipher\;explain\;why$$ I don't think this is true, and assume it is a typo, $K(1,b)$ I would ...
4
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1answer
49 views

Perfect squares formed by two perfect squares like $49$ and $169$.

Let $a$ be a perfect square number whose decimal representation is the concatenation of two perfect squares, for example $49$ (from $4$ and $9$), $169$ (from $16$ and $9$), ... and $4900$, $490000$ ...
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1answer
34 views

How many elements does $\mathcal{P}(A)$ have?

Let $A$ be a set of size fifteen. Let $\mathcal{P}(A)$ denote the power set of $A$, that is the set of all the subsets of $A$. How many elements does $\mathcal{P}(A)$ contain? This is the same as ...
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6answers
931 views

Is there any book/resource which explain the general idea of the proof of Fermat's last theorem?

I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public. I mean, books which is not for mathematicians but for the general ...
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0answers
27 views

Abstract algebra: equivalence relations on vector spaces [on hold]

Let $W$ be a subspace of a vector space $V$ over $\mathbb{R}$. (that is the scalars are assumed to be real numbers ). We say that two vectors $u,v \in V$ are congruent modulo $W$ if $(u-v)\in W$, ...
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1answer
23 views

Question about the Chebyshev Inequality.

Let $p_1 < p_2 <\dots < p_n$ be the $n$ first primes listed in crescent order. Using the Chebyshev Inequality (for $x$ sufficiently large) $$0.92\leq \frac{\pi(x)\log x}{x}\leq 1.11,$$ How ...
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0answers
8 views

Bertrand's Postulate and and Chebyshev Inequality

Let $\theta(x) = \sum_{p\leq x}\log p$ and $\pi(x) = |\{p\leq x:p\text{ is prime}\}|$. Using Abel's formula, one can prof the following $$\pi(x) = \frac{\theta(x)}{\log x} + ...
3
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4answers
47 views

Are all those numbers coprime?

The values of $4m^2+1$ and $4m^2+4m+5$ for $m\geq{1}$ are (resp.) 5,17,37,... and 13,29,53,... Those numbers seem to be all coprime : how to prove it if it is true, please ?
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0answers
17 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
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1answer
80 views

Cut squares from sheet

A rectangular paper sheet of M*N is to be cut down into squares. ...
0
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1answer
16 views

Relation between two numbers.

Let $0 \lt x \lt 1$ and $0 \lt \delta \lt 1$ two real numbers. Can I always find something like $x\lt c\delta^2 \lt 1$, where $c$ is a constant that doesn't involve $\delta$ ?
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0answers
20 views

Find the reflection point $P$

On the real number line, paint red all points that correspond to points of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on ...
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3answers
43 views

Congruences in number theory

I am working on a worksheet on number theory and I have to solve the following congruences: $$7^{128}=n\mod 13$$ Find $n$. And $$28x^2=1\mod37$$ How should I solve these congruences? I have no clue ...
0
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1answer
31 views

$\Im\left ((a+bi)^n\right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*}$

I am trying to connect $a,b,n$ such that $$\Im\left ((a+bi)^n \right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*} \mathrm{or } \; \mathbb{R^*}$$ What I tried was write $(a+bi)^n$ as $\sqrt{a^2+b^2} ...
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0answers
12 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
1
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1answer
19 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
0
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1answer
39 views

Length in time to find the longest range of primes between 2 and a 13 million character digit?

I am trying to run a program that tells me how many prime numbers there are in a range of numbers. I run it in intervals of 10,000 to 100,000. How long would the program take to determine all the ...
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0answers
18 views

Quickie on NT notation

Is there a notation for the set of quadratic residues of an arbitrary natural $n$? I can't seem to find it anywhere on the internet, and it would be very nice if I could use this instead of every time ...
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0answers
35 views

Find the value of polynomial. [duplicate]

If the value of $x$ is $2+2^{\frac23}+2^{\frac13} $ than what is the value of $x^3-6x^2+6x$ ?
2
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2answers
59 views

Can someone help me to find a counter example that shows that $a \equiv b \mod m$ does not imply $(a+b)^m \equiv a^m +b^m \mod m$

Can someone help me to find a counter example that shows that $a \equiv b \mod m$ does not imply $(a+b)^m \equiv a^m +b^m \mod m$. I have tried many different values but I can't seem to find one. I ...
1
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0answers
39 views

Prime numbers that fits in a specific pattern

Any series $\displaystyle \sum_{k=0}^{\infty}a_k2^{-k}$, where $a_k\in\{0,1\}$, converges to some $x\in[0,2]$ and since the sequence $a_n$ is unique for each $x\in[0,2]$ there is an bijection between ...
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0answers
40 views

Dividing a set into three non-empty subsets [on hold]

That is my homework at the beginning of new topic and I was supposed to think about the problem; however, I can't find a strategy to come up with an answer - could you help me somehow? In how many ...
0
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1answer
60 views

How can I prove that there's a unique solution to $3^x - 2^y = 17$?

The solution to $3^x - 2^y = 17$ is $(x,y)=(4,6)$ - easily found with probing. How can I prove that no other solutions exist?
2
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2answers
54 views

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? [on hold]

Is there a way to show that $d(n)$, which counts the number of divisors of $n$ is non-increasing? I'm trying to use the Cauchy condensation test to show that $\sum_{n\ge{2}}\frac{d(n)}{n\log^2n}$ is ...
1
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2answers
44 views

Lucas Numbers Proof $L_n = \alpha^n + \beta^n$

Proof by Induction: Lucas numbers are recursively defined as: $L_n = L_{n-1} + L_{n-2}$ where $L_1 = 1$ and $ L_2 = 3 $for $n \ge 3$ Show that: $L_n = \alpha^n + \beta^n$ for $\alpha = ...
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1answer
34 views

Prove that $M_{p}$ is an ideal of the p-integers

I need to prove that: $M_{p}:=\{ x \in \mathbb{Q}:|x|_{p}<1\}=\{ \frac{a}{b} \in \mathbb{Q}:b\in \mathbb{Z}-p\mathbb{Z},a \in p\mathbb{Z} \}$ Is an ideal of the p-integers and p-integers/ $M_{p}$ ...
2
votes
1answer
51 views

Show that the equality is true

If $f$ is a Completely multiplicative function and $g$ is an arithmetic function such as $g(1) \neq 0$ prove that: $$(f\cdot g)^{-1} = f\cdot g^{-1}$$ Any function with the -1 as exponent is the ...
1
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1answer
23 views

Prove some properties of the $p$-adic norm

I need to prove that the p-adic norm is an absolut value in the rational numbers, by an absolut value in a field I mean a function that goes from $K \to \mathbb{R}_{\ge 0}$ such that: I)$|x|=0 ...
2
votes
1answer
33 views

How to show this equality

If $f$ is a multiplicative function and ¨$n$¨ is a square-free positive integer. Prove that: $$f^{-1}(n) = \lambda(n)\cdot f(n)$$ where $f^{-1}$ is the dirichlet inverse and $\lambda$ is the ...
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votes
1answer
38 views

Prove $\sigma(n)> n+\sqrt n $ [on hold]

Given that $n$ is composite number how to establish $\sigma(n)> n+\sqrt n$ ? here $\sigma(n)$ denotes the sum of all positive divisors of $n$.
0
votes
1answer
24 views

Permutation with atleast n unique characters

I came across this question on Google APAC 2015. I am slightly weak with permutations. The problem goes like this: There is a password. We know the length of the password and the characters used ...
0
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1answer
44 views

Exist an explicit formula to calculate the minimum number of divisions by two that leave a rest < 0.5?

I have a number $x \in \Bbb R/\Bbb Z$ (i.e. any number but entires) and I want to know if exist a explicit formula that evade recursion to calculate the minimum n that $$\frac{x}{2^n}\mod ...
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0answers
29 views

Mobius inversion formula

Let $e$ be a positive natural number, there is the following equality of formal power series ...
2
votes
1answer
24 views

Meaning of congruence notation for Bernoulli Numbers

I am studying Theorem 4(von Staudt's Theorem) in Borevich-Shafarevich's Number Theory(1966)(page 384) which states: Let $p$ be a prime and $m$ an even integer. If $(p-1)\nmid m$, then $B_m$ is ...
0
votes
1answer
42 views

Lagrange's Theorem (number theory)

I am currently doing a proof of Lagrange's Theorem (and smaller related results) for an assignment. I believe I've almost got it done, but I need that push over the edge. First, I need to prove that ...
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votes
1answer
33 views

Function Converts all numbers to even number? [on hold]

Hello Is it possible that a function which equals all numbers to even numbers? But dont use divisible and multiply? For example: f(1)=2 f(2)=8 f(3)=4 . . . f(n)=n.th even number. All ...
1
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1answer
53 views

If Robin's inequality ever fails, are there only finitely many colossally abundant numbers that satisfy it?

Let$\ \sigma(n)$ be the sum-of divisors function, with the divisors raised to$\ 1$. If the Riemann Hypothesis is false, Robin proved there are infinitely many counterexamples to the inequality$$\ ...