Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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9 views

Is there a short proof of the existence of $a$ so that $a$ is a primitive root for infinitely many primes $p$?

After looking for a general answer I found Artins conjecture, and I was happy to see so much is known. However I don't know nearly enough to follow the proof, yet it bothers me I can't prove the ...
0
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1answer
15 views

Ideals of $ord$

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. (a) Show that the only nonzero ideals of $R$ are the ...
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0answers
12 views

Asymptotic sums and big-O notation

Suppose I have to compute the following asymptotic sum ($x\rightarrow\infty$): $$ S(x):=\sum_{n\leq f(x)} O(g(x,n))\;, $$ where the function $g(x,n)$ is non-decreasing in $n$, so that in our case ...
0
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0answers
32 views

Hypothetical proof of Goldbach's conjecture?

Goldbach's conjecture: Every even number greater than 4 is the sum of two prime numbers. An equivalent statement is; For all $n\geq 2$ there exists a number $e(n)$ such that $n-e(n)$ and ...
3
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1answer
17 views

How to see that the prime gaps functions isn't monotonic?

Let $g(n)$ be the distance between the $n$th prime and the next. By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic. Further we know that ...
3
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0answers
26 views

Primes and irreducibles of $\{a+b\sqrt{-2}\ |\ a,b \in \mathbb{Z}\}$

Let $R = \{a+b\sqrt{-2}\ |\ a,b \in \mathbb{Z}\}$ Rational primes $p \geq 3$ of the form $p = a^2 + 2b^2$ factorize in $R$ as a >product of two irreducibles which are not associate. Such ...
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1answer
24 views

Are there any known asymptotics for $\sum_{p\leq x} p$? [duplicate]

As a prospective undergraduate who has really benefited from his time on MSE thus far, i recently learnt that there exists asymptotic approximations for $\sum_{p\leq x} 1, \sum_{p\leq x} p, ...
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0answers
15 views

Is there a proof that $(n-2x)! \times n^{2x-1} > n!$ (where $x$ is a function related to the prime counting function

Is it possible to prove the following? Let $\pi$ be the prime counting function and $A(n)=\pi(2n)-\pi(n)$ $(n-2A(n))! \times n^{2A(n)-1} > n!$
5
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1answer
44 views

Express a real number as a product

Hi guys if I have a number $x \in [1,2)$ is it possible to express such number as: $$x = \prod_{j=0}^{+\infty} (1 + \alpha_j 2^{-j})$$ where each $\alpha_j \in \left\{-1,0,1\right\}$? If yes, how ...
3
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2answers
56 views

Prime counting function; when is it true that $\pi(n) > \pi(2n) -\pi(n)$?

Let $\pi$ be the prime counting function. Under what conditions is it proven true that $\pi(n) > \pi(2n) -\pi(n)$, if at all?
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0answers
47 views

For what values of $a, n$ the number $2^a\cdot 3^n+1$ is prime?

Respected All. Please forgive me as I am unable to understand if it is an off topic. I am stuck in a situation. Change the title if you find inappropriate What I am willing to know is that if there ...
0
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1answer
10 views

Writing a product of transpositions as a 3-cycle.

For the case that we have two transpositions equal to each other, say (a b) (a b) then how can I write the product as a product of 3 cycles?
3
votes
1answer
19 views

Quotient & Remainder of $\frac{a}{bc}$

$a, b$ and $c$ are integers: $a,b,c \in \mathbb{Z}$ $b$ and $c$ are greater than zero: $b>0 \wedge c>0$ $q$ and $r$ are the quotient and remainder of $\frac{a}{b}$: $a=qb+r$ $t$ and $s$ are ...
0
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0answers
28 views

PID and irreducibles

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. (c) Show that the only nonzero ideals of $R$ are ...
2
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1answer
32 views

Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
7
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0answers
24 views

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
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1answer
43 views

Question about 'Knuth's up-arrow notation'

Am I right? $$x\uparrow x=x^x$$ $$x\uparrow\uparrow x=x^{x^x}$$ $$x\uparrow\uparrow\uparrow x=x^{x^{x^x}}$$ $$x\uparrow\uparrow\uparrow\uparrow x=x^{x^{x^{x^x}}}$$ ...
7
votes
1answer
58 views

Given $k$, are there infinitely many $n$ so that $w(n) = w(n+k)$?

$w(n)$ denotes the number of distinct prime factors of $n$. I am wondering if any such result is known.
2
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0answers
23 views

Question on a proof of the euler product of the zeta function

Let $\zeta(s)$ be the Riemann zeta function, then we know it satisfies the Euler product for Re$(s) > 1$, $$ \zeta(s) = \prod_{p} (1 - p^{-s})^{-1}. $$ The proof I read, if I recall correctly, was ...
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2answers
32 views

Generalizing Legendre's Theorem on prime factorization of factorials

From Legendre's Theorem we know $$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$. Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for ...
4
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2answers
49 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
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1answer
41 views

Find a formula for this

I need help. I don't know if it is possible. Example formula that uses English instead of math! $f(x) = 3x$ + all previous values of $f(i)$ with $i$ from $0$ to $x-1$, where $x$ is a positive ...
4
votes
1answer
90 views

Is $\arctan2$ irrational? [duplicate]

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
0
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1answer
34 views

How would you prove $ab|c$ knowing that $a|c$ and $b|c$ and $\gcd(a,b) = 1$? [on hold]

If $a|c$ and $b|c$ and $\gcd(a,b) = 1$, prove that $ab|c$.
2
votes
2answers
18 views

Square Free congruence modulo n

I am trying to show that if $a^n\equiv a\pmod n$ for all integers $a$ that $n$ is square free. I have an idea to start with the contradiction that suppose $n=p^2m$ for some prime $p$, then n does not ...
-3
votes
2answers
31 views

Finding Square Roots

Well, i have a method to find square roots to any number. for Eg To find $\sqrt{58}$ or any number. We have to find which perfect square is place before it. 49 is placed before 58. What we have to do ...
1
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1answer
44 views

How I can prove that for any natural number $n$ such that $30<n$, $\pi(4n-3)<n$?

I need to proove that for any natural number $n>30$: $$\pi(4n-3)<n.$$ In this inequality, $\pi(x):\mathbb{N}\to \mathbb{N}$ is the defined as follows: $$\pi(x):=Card(\lbrace p \ | \ p\leq x\ \ ...
0
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1answer
10 views

Injectivity of idele norm map

Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb ...
1
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3answers
58 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
1
vote
1answer
43 views

Find all $n$ such that $3^{2n+1}-4^{n+1}+6^n$ is prime

I'll try to format my question in a manner such that you can skip (irrelevant) parts. Exercise: Find all natural $n$ such that $3^{2n+1}-4^{n+1}+6^n$ is prime. Motivation: I'm trying to ...
0
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0answers
38 views

How to square a number that got more digits than search results “digits” on Google.

I am implementing the quadratic sieve algorithm. And I got run in unexpected problem. Take a look at those two final steps of the algorithm as described in wiki. Use linear algebra to find a ...
3
votes
2answers
78 views

Product of Primes

Let $\mathbb{P}$ denote the set of prime numbers. How would one evaluate $$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that ...
1
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1answer
64 views

Show by mathematical induction that the gcd(n,n+1) = 1 for every integer n.

By mathematical induction, how would you show $\gcd(n,n+1)= 1$ for every integer $n$? I'm thinking you would start by knowing that some integer, d, would divide both n and n+1. That's all I have so ...
0
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1answer
21 views

Checking if a number is Polygonal

Polygonal numbers are of the form $\cfrac{n^2(s-2)-n(s-4)}{2}$, where $s$ is the number of sides of the polygon and $n$ is to say which one it is (the $n^{th}$ $s$-gonal number) So my question is, ...
5
votes
1answer
86 views

How many solutions are there of the equation $(\cos a)^x+(\sin a)^x=1$, $x \in \mathbb{R}$.

Is there any solution of the equation other than $x=2$? Please help me. Thank you in advance.
2
votes
1answer
36 views

How are the nontrivial zeros of the Riemann zeta function calculated?

The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$ and extends to the ...
0
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0answers
61 views

Find the Summation of Summation [on hold]

An Array A consisting of N integers .We perform the following operation M times: for i = 2 to N: Ai = Ai + Ai-1 We have to find xth element of the array ...
1
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1answer
17 views

Connecting homomorphism in Galois homology using the standard resolution

Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the ...
2
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1answer
52 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
1
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2answers
43 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
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5answers
79 views

Is it accurate to say that multiplication of two integers yields an integer?

I am reading a book in discrete mathematics and it assumes that a multiplication of two integers yields an integer. Although that this book's saying is justifiable since the book is making an ...
0
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2answers
19 views

Showing Modulo Congruence Amongst Prime Divisors (Number Theory)

I'm having trouble figuring out how to show the general existence part of the following problem. Suppose $n\in\{1,2,3...\}$ and $n\equiv 7\mod{10}$. Show that $\exists$ a prime divisor $p$ of $n$ ...
1
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0answers
39 views

Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
2
votes
1answer
22 views

Modular Arithmetic and Greatest Common Divisor.

In my algebraic structures textbook I have come across a tricky question that I am trying to solve which goes as follows: suppose that $d|(a^n-1) $ and $d|(a^m-1)$ where $m,n$ are natural numbers ...
0
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1answer
37 views

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$ It's trivial to see that if $a$ is irrational, we get no solution. Thus $a \in \mathbb{Q} ...
33
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7answers
2k views

$-1$ as the only negative prime.

I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than $1$. This got me thinking about negative prime numbers though, and I soon realized that, for ...
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0answers
39 views

on the least primitive root of a prime

There is an article on the least of primitive root of a prime in this link On the second page you will see ...
2
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1answer
53 views

Homeomorphism between $\mathbb{R}$ and $\mathbb{Q}$ - why does cardinality matter?

When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there. Can someone provides ...
0
votes
2answers
37 views

Intersection of dense sets in $\mathbb{N}$

Let's call $A\subseteq\mathbb{N}$ dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$ Is the intersection of two dense sets dense again? Or does the intersection of two dense ...
2
votes
1answer
39 views

What is the significance of Coleman maps arising in Iwasawa thoery?

I have come across two instances of "Coleman map" Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in ...