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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0
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1answer
12 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 ...
1
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3answers
37 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?
5
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1answer
50 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. ...
1
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4answers
59 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 ...
0
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0answers
12 views

Solving a quadratic congruence (mod p).

Solve x^2 == 6 (mod97). There is an algorithm in my book. Initialization: I1: Determine the integers k,m such that p-1= m2^k, where k>= 1 and m is odd. Then 97 - 1 = 3*2^5, so k = 5 and m = 3. ...
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0answers
20 views

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)
0
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0answers
25 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: ...
1
vote
1answer
35 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 ...
0
votes
1answer
79 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.
0
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1answer
34 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 ...
0
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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 ...
1
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0answers
17 views

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 ...
9
votes
0answers
90 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
1k 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 $x=0.x_1x_2\ldots$ ...
2
votes
0answers
11 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 ...
3
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2answers
20 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 ...
0
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0answers
15 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 ...
0
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1answer
37 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: ...
3
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3answers
34 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
39 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, ...
1
vote
1answer
63 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 ...
0
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1answer
25 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 ...
1
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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 $$\frac{p^n}{n}-\frac{2p^{n/2}}{n}\leq N_n?$$
3
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0answers
43 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 ...
1
vote
2answers
46 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 ...
0
votes
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 ...
5
votes
0answers
27 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 ...
0
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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
11 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 ...
3
votes
1answer
38 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 ...
1
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1answer
103 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 ...
1
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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 ....
0
votes
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 ...
-3
votes
1answer
47 views

Prove that $\mathbb{Q}$ is dense in $\mathbb{Q_{p}}$ and $\mathbb{Z}$ is dense in $\mathbb{Z_{p}}$ [on hold]

I need to prove that $\mathbb{Q}$ is dense in $\mathbb{Q_{p}}$ and $\mathbb{Z}$ is dense in $\mathbb{Z_{p}}$.Can someone help me. Thanks in Advance
6
votes
1answer
70 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...
1
vote
2answers
42 views

How else could we solve the congruence?

How could I solve the congruence $6x \equiv 1 \pmod { 5^4}$? I wanted to use the formula $x_n=\frac{5^4+1}{6}$, but calculating this number, we see that it is not integer. How else could we solve ...
0
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0answers
15 views

Number Theory: Ramification

I am currently trying to figure out the following question regarding ramification. Let K = $\mathbb{Q}(\sqrt{5})$, L = $\mathbb{Q}(\sqrt{7})$, M = $\mathbb{Q}(\sqrt{35})$, and KL = ...
2
votes
0answers
41 views

Whats wrong in this proof of $10$ is a solitary number?

Friendly numbers are two or more natural numbers with a common abundancy, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly ...
1
vote
0answers
21 views

Question about a particular case of the sum of three squares.

I was recently given a math problem by a friend; the challenge was to find a rectangular prism whose side lengths (including diagonals and space diagonal) were all natural numbers. I found that for ...
0
votes
1answer
23 views

Solving a system of modular power congruences

I have to find the $x$ value such that $x^k \equiv a_1 \pmod n$ and $x^q \equiv a_2 \pmod n$, where $k$, $q$, $a_1$, $a_2$ are known constants and $n$ is any number. Is there a method to find $x$ ...
1
vote
1answer
51 views

Using Dirichlet's hyperbola method and Dirichlet's formula

Dirichlet Hyperbola Method. For $x \geq 2$: $$ \sum_{n \leq x} \frac{d(n)}{n} = \frac{1}{2} \log^2 x + 2\gamma \log x + \gamma^2 + O(\frac{\log x}{\sqrt{x}})$$ I know already that the summation of ...
1
vote
0answers
36 views

crazy number of divisors chart on Wikipedia - $\Omega(n)$

I am learning about factorization into primes and I learned about the sum number of divisors function (counting multiplicity). $$ \Omega(n) = \Omega \left(\prod p_i^{a_i}\right) = \sum a_i $$ The ...
15
votes
4answers
1k views

simple quadratic, crazy question

Have been working on this for years. Need a system which proves that there exists a number $C$ which has certain properties. I will give a specific example, but am looking for a system which could ...
6
votes
0answers
40 views

Does this sequence contain all positive integers?

Set $a_1 = 1$. Then $a_n$ is chosen to be the smallest distinct positive integer such that $$\frac{\sum_{i = 1}^n a_i}{n}$$ is a Fibonacci number. If my calculations are correct, the sequence starts ...
0
votes
1answer
36 views

What else could we do, instead of solving the congruences?

I want to find the p-adic expansion of $\frac{1}{p}$ and $\frac{1}{p^r}$ in the field $\mathbb{Q}_p$. So, do I have to solve the congruences $px \equiv 1 \pmod {p^n}, p^r x \equiv 1 \pmod { p^n }, ...
0
votes
0answers
9 views

Root of unity in a global field

Let $K$ be a global field. How to show that $|x|_v= 1$ at every place $v$ of $K$ if and only if $x$ is a root of unity in $K$.
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0answers
28 views

$\Bbb{R}/n\Bbb{Z}$ is isomorphic to $A_\Bbb{Q}/(\Bbb{Q}+C_n)$.

Let $A_\Bbb{Q}$ be the adele group of $\Bbb{Q}$. Let $C_n=\{x \in A_\Bbb{Q}: x_\infty=0 \text{ and }x_p \in p^{\operatorname{ord}_p(n)}\Bbb{Z}_p \text{ for prime }p\}$. I want to show that ...
1
vote
0answers
32 views

Any results for small number Goldbach conjecture research?

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc). My understanding is that those ...
0
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0answers
26 views

Why does Vinogradov method fail to prove even number Goldbach Conjecture? [on hold]

Why does Vinogradov method to prove three prime theory fail to prove even number Goldbach Conjecture ? Will multi-variable Trigonometric Sums be able overcome the difficulties ?