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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Could any one explain the difference between the theorems?

In the paper http://annals.math.princeton.edu/2007/165-2/p04 Theorem 2. Let $b \ge 2$ be an integer. The b-ary expansion of any irrational algebraic number cannot be generated by a finite automaton. ...
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2answers
47 views

Imaginary Numbers

I imagine there have been many questions about imaginary numbers, so if I am asking a question already answered my apologies. I understand that it is perfectly correct to create new number systems ...
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2answers
115 views

How many even numbers are the sum of at most one pair of prime numbers?

Consider the set of all even numbers larger than $2$. Goldbach's conjecture states that every element is the sum of a pair of prime numbers. It has not been proved that all elements abide to that ...
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49 views

Proof that $G(3)\le 7$

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers. Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...
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Problem with the hyperelliptic equation

Suppose $K$ is an algebraic number field with $ [ K : \mathbb{Q} ] = d $. $X, Y , \alpha_1 , \ldots \alpha_n $ are in $O_K$ , i.e. are integral over $\mathbb{Z} $. Suppose that we have the following ...
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What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gower's book, stating that two polynomials with integral coefficients have the same ...
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2answers
27 views

Show that sum of divisors of a composite number $n$ is $> n+ \sqrt{n}$

The hint says to use : When $1\lt d\lt n$, $1 \lt n/d \lt n$. If $d\le \sqrt{n}$, then $n/d \ge \sqrt{n}$ My try : Since there will be atleast half divisors $> \sqrt{n}$, the sum of divisors ...
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1answer
65 views

how to show that the only rational solutions of the equation $x^4+y^4=1$ are $(0,土1), (土1,0)$?

how to show that the only rational solutions of the equation $x^4+y^4=1$ are $(0,土1), (土1,0)$ ? the hint seems like descent argument, but I can't find how to formulate the argument... Can anybody ...
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1answer
53 views

Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
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23 views

Showing $y=\Omega\left(\frac{x}{\log x}\right)$

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $$ \displaystyle \sum_{i=1}^n \dfrac{1}{a_i} \leq 1$$. Let $ y$ denote the number of positive integers smaller that $ x$ ...
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3answers
56 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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2answers
39 views

Mills' constant calculation

How simple method can be calculated with very high precision (hundreds of thousands of decimal places) Mills' constant? http://en.wikipedia.org/wiki/Mills%27_constant
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2answers
43 views

For how many integral value of $x\le{100}$ is $3^x-x^2$ divisible by $5$?

For how many integral value of $x\le{100}$ is $3^x-x^2$ divisible by $5$? I compared $3^x$ and $x^2$ in $\mod {5}$ i found some cycles but didn't get anything
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1answer
33 views

Conditions on solutions of a diophantine equation.

I wanted to list all the natural number solutions $(d_1,d_2,...,d_n)$ to the equation: $$\sum_1^n \frac1{d_i} = 1$$ I could not succeed. I noted that for $n=4$, $(2,4,8,8), (3,3,6,6), (2,3,12,12), ...
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1answer
74 views

how to prove : there are an infinite number of points on the circle

I think the follow problem is equal to the problem set 1.16.(a) in Principles of Mathematical Analysis (walter ruldin), And we take (a, b) in $R^2$, X in $R^i$ how to prove : there are an infinite ...
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3answers
115 views

Infinitude of prime numbers

Everyone knows that there is an infinitude of primes. I know the Euclide, the Euler and the Erdos proofs. But are they the only known proofs ? I will try, here, to present a fourth one : Let the ...
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1answer
44 views

Representation of positive rational numbers as series.

So, in my introductory course on calculus my professor formulated this problem: Prove: Every positive rational number can be written uniquely as: ...
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0answers
89 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1}$ = 1 $\sqrt{1 +\sqrt{4}}$= sq rt of of 3 $\sqrt{1 +\sqrt{4+\sqrt{9}}}$=1.909385061 $\sqrt{1 ...
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0answers
66 views

After how many steps can compositions of $x\mapsto x+1$ and $x\mapsto x^2+1$ produce the same result starting from $1$ and $3$?

This problem is from a Turkish contest: Let $P(x)=x+1$ and $Q(x)=x^2+1$. Consider all sequences $(x_k,y_k)$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k),Q(y_k))$ ...
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0answers
47 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [on hold]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
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67 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum involving $p$-th roots of unity (where $\frac{1}{t}$ is to be understood as the field inverse $t^{-1} \bmod p$ etc.) of the form $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
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2answers
44 views

Adjacent non-coprime numbers

Are there any adjacent pairs of numbers that are not coprime? If so, what are they? If not, is there a proof for this, and what is it. From ordinary thought it seems like there aren't but is there an ...
4
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1answer
33 views

How to bound the following sum

I am interested in bounding the sum $$S(x)=\sum_{i\leq x}\vert\{x/i\}-\{x/(i+1)\}\vert$$ where $\{x\}$ is the fractional part of $x$. A calculation on MATHEMATICA seems to suggest ...
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0answers
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Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$ G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n, $$ what techniques are available for locating, or finding bounds on, the ...
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Binomial Congruence Mod primes

So while I was messing around with binomial coefficients I noticed that $$ \binom{3p-1}{p}\equiv 2 \pmod{p^3} $$ For all the primes I tested above 2. I looked around and found similar congruences ...
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1answer
40 views

looking at the alphabet ,the letters are numbered 1-26 ,

looking at the alphabet ,the letters are numbered 1-26 , such that 1 =one=15+14+5=34 (O=15, N=14, E =5 ) 2=two=20+23+15=58 (T=20, W=23, 0=15) 3=three =56 4=four=60 ...
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0answers
32 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
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2answers
49 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
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0answers
48 views
+50

Differences between large numbers with many factors has little factors

I apologise beforehand for the informality and lack of precision in this question but it is that way because it comes from only an intuition, nothing more than a heuristic argument. Say one has two ...
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0answers
56 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
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1answer
33 views

Order of an integer

Why is it true that: if a has order 3 modulo p then $1+a+a^2 \equiv 0 \, \text{mod}\, p$ Thank you!
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1answer
70 views

An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
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1answer
26 views

Another exercise in number theory

I wanted to ask you to help me with this exercise in numer theory. Here it is: If $g$ is a primitive root modulo $p$ and $d|p-1$, show that $g^{(p-1)/d}$ has order $d$. Show also that $a$ is a ...
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3answers
41 views

Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
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2answers
29 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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Primes made from sequential digits

While messing around, I noticed that across some prime numbers contain only sequentially increasing digits, e.g. $23, 67, 89,23456789$. If we adopt a convention of returning to $1$ after a $9$, we ...
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2answers
78 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
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6answers
626 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
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1answer
26 views

Some questions about sub-fields of the field of complex numbers

Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that (1) $F(f)$ contains $f$ as a sub-field and (2) ...
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Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?

This is a subquestion of this question on MO. Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of: ...
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Prove that for $a>1$ $\Psi_a( n) \neq \infty$ $\forall a, n\in \mathbb{N}$

Let us consider successive powers of an integer $a$ $(>1)$. Let $\Psi_a(n)$ denote the exponent at which $n$ first occurs in the decimal expression. For example $\Psi_2(2)=1$ and $\Psi_2(3)=5$. ...
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3answers
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A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
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1answer
32 views

Is there an upper bound (better than Robin's inequality) to the sum of divisors of non-highly-abundant numbers?

I'm looking for $ f(n) $ such that $ \sigma(n) \le f(n) < ne^\gamma \log \log n $, with $ n $ not highly abundant. I'd like a proof as well. I hope the question is well formatted, I'm posting ...
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2answers
61 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
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Another question/observation about Mersenne numbers and Euler's totient function

This is a follow up to this question Upper bound for Euler's totient function on composite Mersenne numbers and an ongoing project with lots of questions related to Mersenne numbers. I'm sorry if ...
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1answer
34 views

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$? Would you use $\lim_{x\to \infty}\frac{\pi(x)\log(1-\frac{1}{x})}{\frac{1}{\log x}} = 1$? and how would you show this? Can you ...
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0answers
39 views

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$ Can anyone give me some hints on how to solve this? I've tried using logs and get \begin{align} ...
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0answers
45 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
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45 views

What is the highest power of the prime p [on hold]

What is the highest power of the prime p dividing :- a) the product 2.4.6....(2n) of the first n even integer b) the product 1.3.5 ... (2n+1) of the first n odd integers
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30 views

Mersenne numbers with two distinct prime factors

For an integer $k$, denote with $p_k$ the $k$-th prime factor. Let $q$ be an odd prime such that $M_q = 2^q-1$ has exactly two distinct prime factors, say $p_s, p_{s+i}$. What is the largest possible ...