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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2
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1answer
53 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
1
vote
0answers
54 views

Find a closed form for the constant term

In a previous question, an asymptotic expansion was provided for the weighted divisor summatory function $\displaystyle \frac {d(n)}{n}$: $$\sum_{n\leq ...
6
votes
4answers
153 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
0
votes
0answers
33 views

Sum of digits of numbers in a range

Given an integer N. For each pair of integers (L, R), where 1 ≤ L ≤ R ≤ N we can find the number of distinct digits that appear in the decimal representation of at least one of the numbers L L+1 ... ...
0
votes
0answers
35 views

If $f(d)$ is multiplicative, then show that $F(n) = \sum_{d|n} (f(d))^k$ is also multiplicative

If $f(d)$ is multiplicative, then show that $ F(n) = \sum_{d|n} (f(d))^k$ is also multiplicative $n \ge 1$ is a positive integer $d$ is a positive divisor of $n$ $k \ge 1 $ is an integer My try ...
0
votes
0answers
19 views

There exists only a finite number of ideal classes in a number ring

Let $K$ be a number field (i.e. $\mathbb Q\le K\le\mathbb C$ s.t. $[K:\mathbb Q]=n$) and $R=\mathbb A\cap K$ the relative number ring. Calling $\Phi(R)$ the set of ideals of $R$, we define on it the ...
2
votes
1answer
57 views

A Quadratic Diophantine equation in three variables

Can the following Diophantine equation be solved for all solutions? $$x^2 + yz = 2$$ Then get a closed expression for $x_n, y_n, z_n$. It may be related to primes. I'll explain further how I got ...
0
votes
1answer
66 views

Solve $i^3j-j^3i=x^3y-y^3x$

Do anyone have an idea about how to solve this kind of equation: $i^3j-j^3i=x^3y-y^3x$ where $i,j,x,y$ are distinct natural numbers and $i>j$ and $x>y$ Regards
2
votes
0answers
24 views

Number of nonnegative solutions of linear diophantine inequality

Given inequality $Ax + By \le C$, where $A, B, C$ are integers, $A$ and $B$ are coprime and $C < AB$. I need to find number of non-negative integer solutions of it. Is there exists algorithm which ...
1
vote
0answers
47 views

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. ...
1
vote
2answers
62 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 ...
3
votes
2answers
223 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 ...
0
votes
0answers
92 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 ...
2
votes
1answer
23 views

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 ...
1
vote
2answers
32 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 ...
2
votes
1answer
78 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 ...
-1
votes
1answer
63 views

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

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
1
vote
0answers
31 views

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

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$ ...
0
votes
3answers
60 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 ...
0
votes
2answers
41 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
1
vote
2answers
49 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
5
votes
1answer
41 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), ...
3
votes
1answer
81 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 ...
3
votes
3answers
117 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 ...
2
votes
1answer
49 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: ...
2
votes
0answers
135 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 $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ $\sqrt{1 ...
3
votes
0answers
92 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 $(P(x_k),Q(y_k))$ ...
2
votes
0answers
54 views

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

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 ...
5
votes
1answer
147 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 ...
0
votes
2answers
46 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
votes
1answer
37 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 ...
0
votes
0answers
18 views

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 ...
2
votes
0answers
35 views

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 ...
-1
votes
1answer
48 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 ...
3
votes
0answers
40 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 ...
0
votes
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 ...
4
votes
0answers
119 views

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 ...
2
votes
0answers
58 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 ...
0
votes
1answer
37 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!
2
votes
1answer
73 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 ...
1
vote
1answer
28 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 ...
0
votes
3answers
43 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 ...
0
votes
2answers
30 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?
1
vote
0answers
46 views

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 ...
4
votes
2answers
86 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 ...
4
votes
6answers
643 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.
1
vote
1answer
28 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) ...
0
votes
0answers
31 views

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: ...
1
vote
0answers
22 views

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 $\Psi_2(3)=5$. ...
1
vote
3answers
58 views

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.