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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0
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1answer
14 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 ...
0
votes
1answer
22 views

Cut squares from sheet

A rectangular paper sheet of M*N is to be cut down into squares. ...
0
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0answers
3 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} + ...
2
votes
4answers
37 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 ?
1
vote
0answers
18 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 ...
28
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5answers
768 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
1
vote
0answers
14 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 ...
0
votes
1answer
14 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 $\delta$ ?
2
votes
1answer
93 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
0
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3answers
39 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 ...
3
votes
1answer
52 views

Convergence of a series concerning the multiplicative order of 2

I was trying to bound the value of $v_p(2^n-1)$ and some of the series I obtain made me wonder about the following problem. Problem : When does the series $$\sum_{prime \: p} \frac{1}{(ord_p ...
0
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0answers
32 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 ...
0
votes
1answer
43 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 ...
0
votes
1answer
28 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} ...
37
votes
9answers
7k views

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
7
votes
0answers
122 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
1
vote
1answer
18 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 ...
1
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0answers
36 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 ...
0
votes
0answers
13 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 ...
1
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2answers
43 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 = ...
-1
votes
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 $x^3-6x^2+6x$ ?
0
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0answers
25 views

Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
2
votes
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 ...
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votes
0answers
37 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
votes
2answers
171 views
+100

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 $(p,q)$ ...
0
votes
1answer
59 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
votes
2answers
51 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 ...
0
votes
1answer
32 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
2answers
188 views

Solutions of Diophantine equation

Does there exists any other solutions of the following Diophantine equation $$zx^2 +xy^2 +yz^2 =xyzt .$$ I found that $$(x,y,z,t) =(s,s,s,3) ,(x,y,z,t)=(s,2s,4s ,5)$$ where $s\in\mathbb{N}$ are ...
2
votes
1answer
49 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 ...
8
votes
7answers
5k views

The product of n consecutive integers is divisible by n factorial

How can we prove that the product of n consecutive integers is divisible by n factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that ...
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 ...
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 ...
2
votes
3answers
63 views
+100

How amount of Emirp Primes depends on the base of numeral system

There was a problem on searching for primes which, if their decimal notation is reverted, yield another primes, like 37 => 73 or ...
1
vote
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 ...
3
votes
2answers
324 views

How many number square-free integer from 1 to 2013

Question: Let $Q(x)$ denote the number of square-free (quadratfrei) integers between $1$ and $x$ find $Q(2013)=?$ My try:I know $ 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, ...
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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 $n$.
3
votes
2answers
63 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
1
vote
1answer
41 views

Calculation of products of powers using Modular Exponentiation

I need to devise an algorithm that outputs $x^a * y^b$ (mod $m$) on an input of $m, x, y, a, b$ using the binary left to right modular exponentiation algorithm. It should be able to compute $x^{22} * ...
1
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0answers
28 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 ...
1
vote
1answer
51 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$$\ ...
-2
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 ...
5
votes
1answer
84 views

If $a^n-1$ is divisible by $b^n-1$ for all $n$, then $a$ is a power of $b$

Let $a,b$ be natural numbers not equal to $1$ such that $\frac{a^n-1}{b^n-1}$ is natural for any natural $n$. Prove that $a=b^m$ for some natural $m$.
0
votes
1answer
24 views

Minimal distance between coprimes

For a natural number $K$, I want to choose $n$ pairwise coprime numbers all of which are bigger than $K$ such that the distance $d$ between the smallest and the largest one is minimal. For example, ...
28
votes
1answer
655 views

Is there a power of 2 that, written backward, is a power of 5?

In this note the famous mathematical physicists Freeman Dyson gives an example of a true statement that is impossible to prove. Or so he states. The statement is as follow: Numbers that are exact ...
1
vote
1answer
29 views

Does the order of $a$ in $\left(\mathbb{Z}/ \Phi_n(a) \mathbb{Z} \right )^{\times}$ equal $n$?

Here, $\Phi_n(a)$ is the nth cyclotomic polynomial evaluated at a. It's obvious that the multiplicative order of $a$ modulo $\Phi_n(a)$ divides $n$, because $a^n \equiv (a^n-1)+1 \equiv P(a)\cdot ...
2
votes
1answer
29 views

Coin-tossing games

Suppose that you start off with $100$ dollars. You toss a coin $10$ times and guess it right $5$ times and lose $5$ times (the order of the outcomes is not known). It is known that every time you ...
3
votes
2answers
35 views

Infinitely many $n \in \mathbb{N}$ such that $\mu(n) + \mu(n+1) = 0 $.

I try to show that there are infinitely many numbers $n \in \mathbb{N}$ such that $\mu(n) + \mu(n+1) = 0 $. What I did We write $\mathcal{P}$ for the set of prime numbers. We need to show that ...