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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Number theory proofs relating to units

What is a counterexample for the proposition: If u ∈ Um has order n1 and u2 ∈ Um has order n2 then u1*u2 has order [n1,n2] (The LCM of n1 and n2)
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2answers
82 views

How to find the roots of polynomials in $\Bbb Z_p$

Let $p$ be a positive prime. What are the roots of the polynomial $x^{p-1} - 1$ in $\Bbb Z_p$? Factor this polynomial into linear factors in $\Bbb Z_p[x]$. What does this factorization tell you about ...
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0answers
33 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
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2answers
30 views

Proofs regarding units in number theory

I was given this problem to prove and I have no idea how to prove it: How do you prove that if $u \in U_m$ has order $n$ and if $n=kg$, than $u^k$ has order $g$. Any ideas?
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0answers
16 views

On certain error terms

Define $$E_k(x)=-\sum_{d|k}\mu(d)B_1\left(\left\{\frac{x}{d}\right\}\right)$$ where $\mu(d)$ is the mobius function, and $B_1(x)$ is $x-1/2$ and $\{x\}$ is the fractional part of $x$. define $s(k)$ ...
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0answers
16 views

Integer Solution To System of Linear Function

Hello i reposted the question, because the equation title was a bit misleading. [and i could not edit the question] $c,f$ are not known, so it is not a Diophantine equation, it is more like a system ...
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0answers
14 views

products of bernoulli poynomial

prove the following assuming that $d|k$, $e|k$, $(d,e)=gcd(d,e)$, $B_1(x)=x-1/2$, $\{x\}$ is the fractional part of $x$ $$ \int_0^k ...
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1answer
38 views

What are the conditions needed by $n$ so that $2^{n+1}5^n-1$ is a prime number?

What are the conditions needed by $n$ so that $2^{n+1}5^n-1$ is a prime number? I am investigating prime numbers $2^{n+1}5^n-1$ for some $n\in\mathbb{N}$ and I found out that there are prime numbers ...
2
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0answers
35 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
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4answers
56 views

Finding roots of equation in $Z_{14}$, $Z_{17}$

Find all the roots of the equation $x^2 - 9x + 6 = 0$ in $Z_{14}$. Factor the polynomial $x^2 - 9x + 6$ into linear factors in $Z_{14}[x]$ in all possible ways. Find all roots of $x^2 - 9x + 3 = 0$ ...
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2answers
53 views

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
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2answers
23 views

Finding Solutions to a Diophantine Equation with Factorials

How many ordered pairs of positive integers $(a, b)$ are there such that $a!+\dfrac{b!}{a!}$ is a perfect square? Is the number of solutions finite? Source: Ran into it on Facebook. I have plugged ...
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0answers
103 views

Very tight prime bounds

Is it possible that $$\left|\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}\right|<\dfrac{2\sqrt{n}}{e\log(n)}?$$ Since $$ ...
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1answer
25 views

Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
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1answer
149 views

Why elements of the set can be Goldbach pairs for a given even number? [on hold]

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
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1answer
62 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
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0answers
25 views

Number of excellent pairs is equal to $\sigma(n)$

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called good if $$a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu ...
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0answers
996 views

Is $ \lim_{x \to 0} G(e^x) x = \sum_{r=1}^\infty \mu(r) \int_{-\infty}^\infty \frac{G(z^r) dz}{z} $?

I recently derived a formula and was wondering if it was correct or already existing? We define: $$ \sum_{r=1}^\infty \mu(r) G(x^r) = g(x) $$ Where $ \mu(r) $ is the Mobius function. G(x) is an ...
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1answer
45 views

squares that can be divided to two squares

There are some squares like 169 that can be divided into two squares(16 and 9). I classify them into two groups: A:squares that their rightmost number isn't 0(like 169 and 4225) B:squares that their ...
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0answers
41 views

How many units are there in $ \mathbb{Z}[i\sqrt{2}]/2^{2012} $ [on hold]

How many units are there in $ \mathbb{Z}[i\sqrt{2}]/2^{2012} $
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1answer
45 views

Something related to Frobenius coin Problem/Chicken McNugget Theorem

Let positive integers $a,b,c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,y,z$. Prove that $$ g(a, b, c)\ge ...
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0answers
38 views

Transcendental number, relative to a set of functions [on hold]

Whatever the finite set of functions FOX = {F1, F2, ... Fn}, where Fi is defined and continuous on the interval J | R; i = 1,2,3 ..., n, .... etc. While there are countless real numbers in J as x, ...
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0answers
17 views

Solve the generalization of the Erdős-Straus conjecture, but one of a, b, and c negative?

http://cjoint.com/?DGljiVok1rl It was my personal traveaux, I would like to know your opinions, please
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0answers
23 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
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0answers
19 views

Multiplicative function in Number Theory [on hold]

Let $f(x)$ be a polynomial with integer coefficients, and let $w(n)$ be the number of $f(j), j=1,2,3,...,n$ such that $\gcd(f(j),n)=1$. Show that $w(n)$ is multiplicative.
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1answer
49 views

Has anyone solved this general Diophantine Equation?

I know that Pythagorean triples have been parameterized, I also know that Andrew Wiles has proved that there are no distinct integer solutions for $ a^n + b^n = c^n$, when $ n \ge 3 $. However we may ...
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1answer
32 views

Normal number generator with digit extraction algorithm?

Are there any known ways to define an absolutely normal number (or very likely normal) number, which posses digits that can be extract via algorithm? I want to find numbers like pi that are normal and ...
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4answers
104 views

sum of two consecutive squares that add up to a square

Find all integer $x$ s.t. there is an integer $r$. $x^2+(x+1)^2=r^2$ What if $x$ is a square? All of my attempts lead to dead ends.
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2answers
46 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
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2answers
44 views

Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
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2answers
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Does the absence of horizontal lines shows that there are no $n,m\in \mathbb{N}$ such that $n^2=2m^2$?

When I was learning about the proof of the irracionality of $\sqrt{2}$, I remember of trying to visualize it by ploting the graphs of $f(n)=n^2$ and $g(m)=2m^2$, but at the time I got confused and ...
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0answers
265 views

What is the efficient way to calculate number of divisors of N that are divisible by 2?. [on hold]

Moderator Note: This has been claimed to be a current contest question on hackerrank. For example if a number is given let say 8 then its factors are 1,2,4,8 hence total numbers of divisors which ...
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1answer
62 views

Is there a prime between $k$ and $\dfrac{11}{9}k$, $\forall k\ge 24$?

Given $k\in\mathbb{N}$, $k\ge 24$, is there always a prime number in the interval $\left[k,\dfrac{11}{9}k\right]$? I tried to verify this statement with the computer and it seems to hold. Is it ...
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0answers
53 views

Sum of possible permutations

Lets call two arrays A and B with length n almost equal if for every i (1 <= i <= n) CA(A[i]) = CB(B[i]). CX[x] equal to number of index j (1 <=j <= n) such that X[j] < x. For two ...
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0answers
161 views

Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
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2answers
79 views

Fermat's Last Theorem for Negative $n$

While studying Fermat's Last Theorem and Pythagorean triples, the following question occurred to me: For the equation $a^n+b^n=c^n$, where $n$ is a negative integer, a) does a solution exist, and b) ...
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0answers
22 views

What is expected value of multiplication of two numbers iteratively?

Given some numbers [1..n]. given some intervals [Li..Ri]. and given value space of random variable. Every-time we can choose some value from value-space, and multiply with numbers in interval. How ...
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1answer
46 views

Other Interesting solutions to $a=bq+r$? [on hold]

The division algorithm says $a=bq+r$, with $r$ between $0$ and $b$. Are there interesting restrictions on $r$ using number-theoretic properties that make the equation $a=bq+r$ hold, or hold with ...
2
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1answer
73 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
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1answer
53 views

Diophantine equations in $\mathbb{Z}[i]$ [on hold]

Please help me solve the following question: Solve the linear Diophantine equation in $\mathbb{Z}[i]$: $$(3+5i)x + (5+2i)y = 1$$ Thanks!
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1answer
27 views

Definition of p-adic neighborhood [on hold]

I have some information about p-adic metric. I wonder if anybody has information about the p-adic neighborhood subject? Do they have the same meaning?
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2answers
1k views

Do 4 consecutive primes always form a polygon?

Related to this question, if 4 segments have length of 4 consecutive primes, can they always form a 4-vertex polygon? This question occurred to me out of sheer curiosity, but now I can't prove or ...
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1answer
27 views

Solving a Quadratic Diophantine equation for three variables

I am trying to solve the following Diophantine equation $13x^2-y^2=z^2$. Is there a standard method for generating x and y so that $13x^2-y^2$ is always a square? Mathematica gives me the following ...
2
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1answer
101 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
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1answer
48 views

Understanding Hensel's Lemma

I am learning Hensel's Lemma and trying to solve the polynomial congruence $$x^5+x^4+1\equiv 0\pmod{81}$$ Now my professor taught us the technique of building up from $p$, to $p^2$, and continuing to ...
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2answers
31 views

An exercise in number theory: associates elements

I have a question for you about associates elements in an integral domain. Let $R$ be an integral domain and define $aR := \{ ar \; | \; r \in R\}$. In the following, for $unity$ (denoted with $u$) I ...
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0answers
60 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
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2answers
46 views

Solving $x^2+y^2=a$

I'm looking for a lesson or a link to a well explained proof on how to solve this kind of equations: $x^2+y^2=a$ where $x,y,a$ are all naturals. Regards
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0answers
27 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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1answer
160 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?