Tagged Questions

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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Goldbach conjecture / RH

Would the proof of the strong form of the Goldbach conjecture imply the Riemann Hypothesis? I know the reverse is true for the weak Goldbach conjecture, but I have found nothing regarding the former.
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3answers
39 views

Finding the minimum number which can be divided by $(2, 3, 4, 5, 6, …1000)$ without remaining?

If I want to find the minimum number which can be divided by 2, 3, 4...10 without remaining, the solution will be N=2520 But If the range of numbers become very wide, how can I find this minimum ...
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13 views

Congruence and Legendre symbol problem

Let $p,q$ different odd prime numbers and $a$ a positive odd number, Prove that : $$p \equiv - q\ \ \ \ (mod \ \ 4a )\Longrightarrow \Bigg(\frac{a}{p}\Bigg) =\Bigg(\frac{a}{q}\Bigg) $$ Where ...
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1answer
11 views

How to show this Legendre Symbol Problem

Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$, Prove that: $\exists x, y \in \mathbb{Z};\,\, \gcd(x,y) = 1$ such as $x^{2} +ny^{2} \equiv 0\, (\mod p) \iff ...
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20 views

How to choose $a, b, c$ such that $gcd(a, b, c)$ | $d$ or $gcd(a, b)$ | $d$ or $gcd(b, c)$ | $d$ or $gcd(a, c)$ | $d$

Find the number of ways in which $3$ distinct integers $a, b, c$ $(1 \le a, b, c \le 100 )$ such that $gcd(a, b)$ or $gcd(b, c)$ or $gcd(c, a)$ or $gcd(a, b, c)$ divides $d$, where $d$ is any integer. ...
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1answer
21 views

Periodocity of $a^{pn+q}$ mod $m$

Is $a^{pn+q}$ mod $m$ periodic? $a$, $p$ and $q$ are constants. $n$ is varied here. If it is periodic then how can I find the periodicity efficiently? Thanks in advance.
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1answer
24 views

Question about Number Theoretic Function

I know how to prove (a). It is just $2^w(mn)=2^w (m) \cdot 2^w (n)$. However, for part (b), I can just write $n$ as $p_1^{k_1}\cdot p_2^{k_2} \cdots p_r^{k_r}$, or write d as this kind of form, ...
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2answers
32 views

Let $n$ be a 3-digit number. Prove $9\mid n$ iff the digits of $n$ sum to a multiple of 9. [duplicate]

I have convinced myself that this true, however I'm at a loss of where I should start with this proof. Looking at a similar proof with 3 instead of 9, I saw the use of the basis representation ...
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1answer
65 views

Prime number conjecture

It was suggested that I put my full conjecture up instead of specific examples. Here it is: Given any prime p>3, there exists c such that the following conditions hold: 1a. The quadratic equation ...
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Is it possible to define the Kronecker symbol $\left(\frac{a}{2}\right)$ using some sort of arithmetic in a manner similar to the Legendre symbol?

And if so, what does it say about the cases some mathematicians prefer to leave undefined? As you all well know, $$\left(\frac{a}{p}\right) = a^{\frac{p - 1}{2}} \pmod{p}$$ for $p$ an odd prime. This ...
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0answers
22 views

How to find an expression for a number via a given set of math operators and constants?

Consider one gives you $n$ digits of a (possibly irrational) number and a finite set of math operators and constants, like $\{\sin(x), \ln(x),+,\times, \pi,e,7\}$. Then he asks you to build a ...
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1answer
29 views

$a_n=2^n+3^n+6^n-1$. Find all positive integers that are primes to all terms of the sequence.

Let the sequence $a_n=2^n+3^n+6^n-1, n\in\mathbb N_{> 0}$. Find all positive integers that are prime to all terms of this sequence. I have no idea how to approach this, but I know that I CAN'T use ...
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1answer
37 views

Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID

How would one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a principal ideal domain (PID)? It isn't a Euclidean domain according to the Wikipedia article on PIDs.
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2answers
25 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid ...
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1answer
14 views

Do the properties of the euclidean norm hold for hyperreals?

I just want to confirm that triangle inequality still holds if I replace the square root function, with it's hyperreal equivalent in the following function: $\rho(x,y) = \sqrt{\sum\limits_{i=1}^n ...
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2answers
71 views

Does $x^2 + x + 1 \equiv 0 \mod p$ have a solution?

Problem: I am trying to prove that $$ x^2 + x + 1 \equiv 0 \mod p $$ has a solution where $p$ is a prime such that $p \equiv 1 \mod 3$, without using quadratic reciprocity. I am also suspecting that ...
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2answers
20 views

Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?

The wikipedia article on p-adic numbers warns about $b$-adic expansions where $b$ is not a prime: Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with ...
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3answers
44 views

$\varphi(n) \leq 5$, where $\varphi$ is the Eulerian function

If $\varphi(n) \leq 5$, then can we find a bound for $n$ itself, where $\varphi$ is the Eulerian function?
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1answer
68 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
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4answers
85 views

How is it true that zero is neither a positive number nor a negative number?

At first, the number zero looked like it was positive to me because positive numbers can be written with or without a plus sign to the left of them, but it's false. I was surprised when I heard that ...
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1answer
18 views

Solving a quadratic congruence (mod p).

Solve $x^2 \equiv 6 (mod~97)$. There is an algorithm in my book. Initialization: I1: Determine the integers $k,m$ such that $p-1= m \cdot 2^k$, where $k \geq 1$ and m is odd. Then $97 - 1 = 3 ...
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Factoring polynomials into irreducible factors [on hold]

Factor the two polynomials (x^8 - 1) and (x^6 - 1) into irreducible factors over a)F[2] b)F[3] c) C (complex numbers)
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27 views

Triangular number puzzle with big numbers

Let $n_T$ be the $n^{th}$ triangular number, 1+2+3+...+n or $\sum_{i=1}^n i$ , which equals ${n(n+1) \over 2}$ . Show there exists some positive integers m and c, such that the following are true: ...
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1answer
37 views

A question about the product of primes

Let $\mathbb{P}$ be the set of all primes in the natural numbers and let $p_i \in \mathbb{P}$ be the $i$th prime, $p_1=2$. Let $m = \prod_{i=1}^n (p_i)$. How many solutions does $x^2 + x \equiv 0 ...
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1answer
86 views

Prove that $n$ doesn't divide $3^n+1$

Prove that any odd integer $n>1$ Doesn't divide $3^n+1$ I know that Fermat's little theorem will be useful in this problem but have no idea how to prove it.
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1answer
39 views

A question on the prime number theorem as presented in the following paper

In the section 2. of this paper it is written that, ...The prime number theorem ensures that we can choose $B$ as close to $1$ as we want, provided $x_0$ is sufficiently large. I think that ...
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1answer
20 views

Range of the given parameters

Let $x, y, z$ be positive real numbers satisfying $\frac13\le xy+ yz+ zx \le 3$. Determine the range of values for (i) $xyz$, and (ii) $x+y+z$. Using the concept of AM greater than GM, which indeed ...
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0answers
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Number with given digit distribution multiplied by rational

Let $x$ be an irrational number whose base-$m$ expansion has each digit $i$ (for $0 \leq i \leq m-1$) appearing with average frequency tending to $p_i>0$ (and $\sum_{i=0}^{m-1} p_i=1$). If $q ...
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0answers
93 views

Prove that $ x^xy^y=z^z $ has infinite integral solutions [duplicate]

Show that there exist an infinite number of solutions for $$ x^xy^y=z^z $$ where $x,y,z \gt 1$ & $x,y,z\in \mathbb Z$ I don't know how to even start,infact I am not able to find a particular ...
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0answers
5 views

$R_S (=K \cap A_{K,S})$ is a Dedekind domain

Let $K$ be a global field and let $S$ be a finite, nonempty set of places of $K$ containing the infinite ones. Show that $R_S (=K \cap A_{K,S})$, the ring of $ S-$ integers of $K$, is a Dedekind ...
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5answers
2k views

Given any computable number, is there any algorithm to decide whether it is transcendental?

Given any computable number $a_c$, is there any algorithm to decide whether it is transcendental? Definition of “computable number”: According to Ming Li and Vitanyi, a real number $x=0.x_1x_2\ldots$ ...
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0answers
12 views

Extension of Zolotarev's proof of quadratic reciprocity

I recently came across the wonderful proof of quadratic reciprocity given by Zolotarev in the $1800$'s and have seen a wonderful visualisation of this given by a "card trick" (see Jerry Shurman's nice ...
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2answers
21 views

Division with remainder

I have proved the division with remainder theorem: If a $\in \mathbb{Z}$ and $d \in \mathbb{N}$ then there exists unique numbers $q,r \in \mathbb{Z}$ such that $a=dq+r$ where $0\le r<d$. I proved ...
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0answers
16 views

Odd length repetends in recurring “decimals”

For any number n the reciprocal can be expressed as a decimal, which will be composed of a recurring pattern as long as n is co-prime with 2 and 5. In general terms 1/n will produce a recurring ...
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1answer
40 views

any computational analytic number theory book?

All: Can anyone recommend an introduction computational analytic number theory book ? I am mainly interested in using computer software to verify and confirm analytic number theorem, things related: ...
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3answers
40 views

Prove that $6|(n^2 - 1)$ if $gcd(6,n) = 1$

I'm working through the problems in this book: Number Theory (Dover Books on Mathematics) and I came across this problem (title). here is my working $gcd(6,n) = 1 \implies 1 = nx + 6y$ for some ...
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2answers
41 views

Does almost every whole number integer contain any of the digits zero through nine?

For example, how many whole numbers contain an eight? Well, for whole numbers less than ten, it's just eight itself, so that's 10% and for whole numbers less than 100, there are 8, 18, 28, 38, 48, ...
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1answer
67 views

who, by doing what, can make major contributions (breakthrough/discoveries) in math research?

I am a Math Ph.D student, had already published two small articles. I want to ask more experienced mathematician a question. What kind of person, by doing what, can make major contributions ...
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1answer
26 views

General Behavior of Euler Totient Function

If we have two integer M and N such that $$GCD(M,N) = k$$ Then what is $$\phi(MN)$$ There is a famous identity which states: $$GCD(M,N)= 1 \rightarrow \phi (MN) = \phi(M)\phi(N)$$ And now I am ...
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1answer
20 views

Easy (?) estimation about prime powers

Let $N_k$ be some integers with $\sum_{k\mid n}kN_k=p^n$. How can I prove $$\frac{p^n}{n}-\frac{2p^{n/2}}{n}\leq N_n?$$
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0answers
46 views

How many numbers are products of $p^p$?

Consider the set $\mathcal{S}=\{1,4,16,27,\ldots\}$ of numbers which are products of numbers of the form $p^p$ for $p$ prime. ($\mathcal{S}$ is A072873 in the OEIS.) Note that multiple primes are ...
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2answers
47 views

Is such a number necessarily irrational?

Suppose $(q_{n})_{n\in\mathbb{Z}_{\gt 0}}$ is a decreasing sequence of positive rational numbers such that $Q:=\displaystyle{\sum_{n>0}q_{n}}$ is finite. Let's denote by $n_{i}$ and $d_{i}$ the ...
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1answer
15 views

Converting to Power of Ten Representaion

For very large calculations Wolfram Alpha offers a variety of different representations of the number. One of these is the number written in the form $10^{10^n}$, where $n$ is usually some long ...
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0answers
33 views

How are Hilbert Space methods used in number theory?

In N. Young's book "An introduction to Hilbert Space," there is an interlude in which the author remarks that the theory of Hilbert spaces is "routinely used in differential geometry, complex ...
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1answer
16 views

Decimal expansion__Represent numbers as $x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$?

If $b>1$ is an integer, is well know that the numbers $x\in (0,1]$, can be written as $$x = \sum_{k=1}^{\infty} \frac{a_k}{b^k}$$ for some integers $a_k \in \{0,1,\ldots ,b-1\} $. My problem is ...
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0answers
12 views

Show that a Field Extension is Unramified Using Transitivity

Let $K=\mathbb{Q(\sqrt{5})}, L=\mathbb{Q(\sqrt{7})}, M=\mathbb{Q(\sqrt{35})}$, and $KL=\mathbb{Q(\sqrt{5},\sqrt{7})}$. Show that $KL/M$ is unramified (i.e. every prime ideal of $M$ is unramified in ...
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1answer
43 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
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2answers
138 views

Simple quadratic, crazy question part 2

In my previous question, I asked for advice on a general method to solve a specific problem. Many good ideas came from this, but the problem I gave was too simple and these approaches were sufficient ...
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1answer
54 views

To prove $\pi(x)>\dfrac x{\ln x} , \forall x \ge 17$ by elementary argument

Is there an elementary argument for proving $$\forall x \ge 17:\pi(x)>\dfrac x{\ln x} $$ ? where $\pi(x)$ is the prime counting function ....
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2answers
40 views

Calculate the power series

I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$. In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$? If ...