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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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 ...
0
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3answers
49 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.
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5answers
1k views
+50

prime as sum of three numbers whose product is a cube

Good evening! I am very new to this site. I would like to put the following materiel from Prof. Gandhi's note book and my observations. Of course it is little long with more questions. But, with good ...
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3answers
102 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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4answers
343 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
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1answer
26 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
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1answer
66 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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0answers
47 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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1answer
40 views

Number theory, Find $n$

Find $n$ such that $n$ divides $2^n + 2$. Also, $n$ should be between $100$ and $1000$. It can be easily seen that $n$ is not a multiple of $4$. By brute force I have figured out that answer is ...
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0answers
56 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 $\sqrt{1 +\sqrt{4+\sqrt{9}}}$=1.99385061 $\sqrt{1 ...
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0answers
20 views

Arithmetic Progression of Modular Solutions

Suppose we have integers $y<x$ and $c<x$ such that there exists a solution $z$ to the modular equation $$xz\equiv c\pmod{x+y}$$ Then the solutions $z_n$ to the series of equations $$xz_n\equiv ...
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1answer
40 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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1answer
168 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 ...
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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
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1answer
43 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 : ...
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0answers
50 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
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1answer
340 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 ...
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2answers
40 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
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0answers
28 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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0answers
43 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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0answers
52 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 ...
4
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1answer
32 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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2answers
25 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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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 ...
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0answers
11 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 ...
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3answers
37 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
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2answers
44 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 ...
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1answer
38 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
26 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 ...
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2answers
48 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
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0answers
25 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
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0answers
55 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
32 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
25 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
72 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 ...
1
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0answers
36 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 ...
8
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3answers
298 views

Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$

I am trying to find all solutions to (1) $y^3 = x^2 + x + 1$, where $x,y$ are integers $> 1$ I have attempted to do this using...I think they are called 'quadratic integers'. It would be ...
4
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6answers
623 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
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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) ...
8
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2answers
150 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
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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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0answers
25 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: ...
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3answers
93 views

Irrational number “test”?

Suppose we have a finite quantity $a$, which we would like to prove to be irrational, supposing that it is indeed irrational. Then, would it be enough to show that ...
4
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3answers
102 views

Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k $$ where $k$ is a integer. I can't just come up with a solution.
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0answers
14 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 $\Psi_2(3)=5$. ...
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2answers
59 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 ...
25
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2answers
491 views

Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit ...
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0answers
23 views

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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0answers
105 views
+50

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
0
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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 ...