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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1
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1answer
13 views

convergence of the sequence $10^{-n}$ in the p-adic numbers

Let $p$ be prime. I am tasked to prove that the sequence $10^{-n}$ does not converge in $\mathbb{Q}_{p}$ for any $p$ where $\mathbb{Q}_{p}$ is the set of p-adic numbers. For $p=2$ or $5$, we see ...
0
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0answers
8 views

Any 2D or 3D extension of Hardy Littlewood Circle Method?

All: Any 2D or 3D extension of Hardy Littlewood Circle Method ? Also any 2D or 3D extension of Vinogradov Trignometic Sum methods ?
1
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1answer
30 views

Equation $x^2 + y^2 + 1 = 0$ (mod $p$)

How to prove that equation $x^2 + y^2 + 1 = 0$ (mod $p$) has roots? Hints are acceptable.
0
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0answers
14 views

Prove or disprove the inequality on $p_n$

If $p_n$ denotes the $n$-th prime then is it true that, $$1>\left(\dfrac{p_{n+1}}{\ln p_{n+1}}-\dfrac{p_n}{\ln p_n}\right)$$ for all sufficiently large $n$. It seems that the inequality is very ...
0
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0answers
14 views

How prove there exsit infinite number of $(m,n)$ such $[n\sqrt{p}]=\dfrac{3m^2-m}{2}$

show that: for any prime number $p$, there exsit infinite number of positive integers to $(m, n)$ such $$[n\sqrt{p}]=\dfrac{3m^2-m}{2}$$ where $[x]$ is the largest integer not ...
2
votes
1answer
11 views

Determinant value of a square matrix whose each entry is the g.c.d. of row and column position

Let $A=(a_{ij})$ be a $n \times n$ matrix with $a_{ij}=\gcd(i,j) , \forall i,j=1,2, \cdots, n$ , then how do we prove $\det A=\prod_{i=1}^n \phi(i)$ ? , where $\phi$ is the Euler's phi function
0
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0answers
18 views

If $(a,b)=\delta_1$ and $(b,c)=\delta_2$ then $(b,ac)=\delta_1\delta_2$

The when both $\delta_1,\delta_2=1$ Now to prove this we use the following : $ap+bq=\delta_1,bs+rc=\delta_2$ for $p,q,r,s\in \mathbb{Z}$ $\therefore$ ...
-4
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2answers
31 views

Euler phi function

How can I Verify that Euler’s phi function gives a result of 40 when applied to the number 75. Extremely confused please help. I was thinking to use the formula of φ(m*n)=φ(m)*φ(n) or may be I should ...
-1
votes
1answer
20 views

Polynomials over a finite field - avaerage value [on hold]

Need to prove that if $$ \lim_{n\rightarrow\infty}\frac{1}{q^{n}}\sum_{\begin{array}[t]{c} f\textrm{ monic}\\ \deg(f)=n \end{array}}h(f) $$ exists, it's equal to: $$ ...
0
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0answers
15 views

Problem on solving congruence equation

I want to find $x$ in this equation for known positive integer values $a$, $b$ and $m$. $$a-bx+2^{6x+1}\equiv0\ \pmod{m}$$ Any help will be appreciated.
1
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3answers
62 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 ...
0
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0answers
15 views

Congruence and Legendre symbol problem [duplicate]

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 ...
0
votes
1answer
14 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 ...
0
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0answers
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. ...
0
votes
1answer
26 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.
0
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1answer
25 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, ...
1
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2answers
33 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 ...
6
votes
1answer
75 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 ...
2
votes
2answers
17 views

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 ...
3
votes
1answer
30 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 ...
2
votes
1answer
31 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 ...
4
votes
1answer
38 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.
1
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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 ...
0
votes
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 ...
1
vote
2answers
72 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 ...
0
votes
2answers
26 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 ...
2
votes
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?
7
votes
2answers
103 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. ...
4
votes
4answers
96 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
votes
1answer
19 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 ...
0
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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
28 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
38 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
87 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
40 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
votes
1answer
21 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
vote
0answers
18 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
94 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 ...
0
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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 ...
18
votes
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 $x=0.x_1x_2\ldots$ ...
2
votes
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 ...
3
votes
2answers
22 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
votes
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 ...
0
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1answer
43 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
votes
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 ...
1
vote
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, ...
0
votes
1answer
70 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
votes
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 ...
1
vote
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 $$\frac{p^n}{n}-\frac{2p^{n/2}}{n}\leq N_n?$$
4
votes
0answers
58 views
+50

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 ...