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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-3
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0answers
27 views

Maths: Profit and Lose. [closed]

A temple has 100 thousand visitors every Friday & the statistics shows that for every 3 new visitors, there are 2 regular visitors. The shop in the temple recruits consultant to increase the ...
5
votes
1answer
115 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 ...
2
votes
0answers
130 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 ...
2
votes
0answers
21 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 ...
4
votes
1answer
260 views

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
0
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1answer
37 views

Can you derive a formula for the semiprime counting function from the prime number theorem?

E.g., there are $4$ semiprimes less than or equal to $10$ $(4, 6, 9, 10)$ or $2$ squarefree semiprimes ($6$ and $10$). It's ok if it's off for small numbers but gets more accurate as $n \to \infty$.
0
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1answer
225 views

How to determine whether a number can be written as a sum of two squares?

I know the following theorems: A number can be represented as a sum of two squares precisely when $N$ is of the form $n^2 \prod p_i$ where each $p_i$ is a prime congruent to 1 mod 4 If the equation ...
1
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0answers
31 views

Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
1
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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
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2answers
60 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 ...
0
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3answers
59 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 ...
3
votes
2answers
214 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
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0answers
86 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 ...
4
votes
0answers
114 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 ...
3
votes
1answer
74 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
vote
2answers
29 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 ...
-1
votes
1answer
59 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
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0answers
30 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$ ...
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 ...
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
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2answers
48 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
12
votes
1answer
343 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
10
votes
2answers
203 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
1
vote
2answers
389 views

twin prime conjecture

Whether I am correct or wrong I don't know. If there are any corrections, please let me know. Let $p_n$ = product of all primes. (of course we can go still beyond as we know $p_n$ is infinite). Now ...
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.
3
votes
3answers
116 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 ...
5
votes
4answers
364 views

Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.

$$p+1=2n^2$$$$p^2+1=2m^2$$ Find positive integers $m,n$ and prime $p$ satisfying the above two equations. What would people commonly do? Subtracting both the equations. You get: ...
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
75 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 ...
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: ...
1
vote
1answer
171 views

Why isn't $1$ a superior highly composite number?

A superior highly composite number is a positive integer $n$ for which there is an $\epsilon>0$ such that $\dfrac{d(n)}{n^\epsilon} \geq \dfrac{d(k)}{k^\epsilon}$ for all $k>1$, where the ...
1
vote
3answers
141 views

How to prove $~(c - b) ^ 2 + 3cb = x^3~$ has no nonzero integer solutions?

I'm trying solve: $~a^3 + b^3 = c^3~$ has no nonzero integer solutions. Only one problem left: because $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = a ^ 3,\quad (1)$ if $~c-b~$ is a cubic number, ...
2
votes
1answer
50 views

Is every positive integer the sum of at most 8 pentatope numbers?

Is every positive integer the sum of at most 8 pentatope numbers ? See : http://en.wikipedia.org/wiki/Pentatope_number I saw this conjecture here : ...
3
votes
0answers
84 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 $(P(x_k),Q(y_k))$ ...
3
votes
1answer
341 views

Definition of a peculiar quotient group of isometries

I just wrote a text file to sum up my ideas about the Riemann Hypothesis. This text is a draft, and I don't expect people here to say if this approach is interesting or not (but if by chance you think ...
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 ...
3
votes
0answers
38 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 ...
2
votes
0answers
52 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 ...
4
votes
1answer
35 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
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?
6
votes
1answer
269 views

Does the inverse of this matrix of size $n \times n$ approach the zero-matrix in the limit as $\small n \to \infty$?

Fiddling with another (older) question here I constructed an example-matrix of the type $\small M_n: m_{n:r,c} = {1 \over (1+r)^c } \quad \text{ for } r,c=0 \ldots n-1 $ . I considered the inverse ...
0
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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
45 views

Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$

From research completely unrelated to Number Theory I stumbled onto the following equation: $$ xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3 $$ for $x, y, z$ integers, $x,y,z \neq 0$(I ...
-1
votes
1answer
42 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 ...
2
votes
0answers
31 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 ...
0
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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 ...
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
34 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!
1
vote
1answer
27 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 ...
4
votes
2answers
84 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 ...