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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0
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17 views

Can this number be rational?

Let $K = e^{-\gamma}\log\log n$, where $\gamma$ is the Euler-Mascheroni constant and $n\geq 2$ is a positive integer. Can $K$ be rational for any integer $n\geq 2$ ? I seem not to find any argument ...
0
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0answers
3 views

Density of diophantine equations for which the local global principle is valid.

(sorry for possible errors) The local global principle says that in some cases a diophantine equation is solvable over the rationals, if it can be solved over the rationals and in p-adic fields $Q_p$ ...
2
votes
1answer
18 views

Show that $g^p (1 + p)$ is a primitive root modulo $p^e$

Given that g is a primitive root modulo $p$, show that $g^p (1 + p)$ is a primitive root modulo $p^e$. I'm not really sure where to go with this. the $ gcd(p^e, g^p (1 + p))$ is easy enough to show ...
1
vote
1answer
28 views

Show that the only nonzero ideals of R are the principal ideals $\langle p^e \rangle$

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ \textrm{ord}_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. Show that the only nonzero ideals of $R$ are the ...
0
votes
1answer
29 views

Solving a quartic congruence modulo 175

The congruence I'm trying to solve is $x^4 \equiv 71 \pmod{175}$. I really have no idea how to approach this as 175 isn't a power of a prime.
2
votes
2answers
25 views

On the relationship between $\phi(n)$ and $\sigma( n)$

I recently learnt that $\frac{\sigma(n)}{n} \leq \frac{n}{\phi(n)}$, were $\sigma(n)$ denotes the divisor function, $\phi(n)$ the Euler totient function and $n\geq 2$ is an integer. My questions is ...
2
votes
0answers
18 views

Choosing three integers to satisfy an equation under a specific condition

Find three integers $(a,b,c)$ such that: $x*a + y*b + z*c = a + b$ only when $x = 1, y = 1, z = 0$ where $x, y$ and $z$ can be chosen as any non-negative integers. For example, choosing $a = 1$; $b = ...
0
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0answers
23 views

Regarding the iteration of sum of prime factors

Let $sopf(n)$ be the sum of prime factors of $n$, with repetition for prime factors. I have observed an interesting phenomena when $n$ is a prime number $p$. So for any prime number $p_1$, if ...
0
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0answers
40 views

Test if a number is in ${\mathbb R}$ [on hold]

Given a number $x$ $\in$ ${\mathbb R}$ is there a way to know if $x$ $\in$ ${\mathbb N}$ without comparing $x$ with a number in any known list of numbers? (ex. {0,1,2,3,...}) To be more specific: ...
8
votes
4answers
353 views

Decimals of the square root of $n$.

Let $a_1, \ldots, a_k$ be any sequence of digits (i.e., each $a_i$ is between 0 and 9). Prove that there exists an integer $n$ such that $\sqrt{n}$ has its first $k$ decimals after the decimal point ...
4
votes
0answers
24 views

Proving the congruence of a Fibonacci Number [on hold]

Let $F_n$ denote the $n^{th}$ fibonacci number where $F_0 = 0, F_1 = 1$. Prove that for all primes $p > 5$, $$F_p \equiv 5^{\frac{p-1}{2}} \mod (p)$$
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votes
2answers
30 views

To find composite integers satisfying the given property.

Find all positive composite integers $n$ greater than $1$ such that for any relatively prime divisors $a$ and $b$ of $n$ with $a > 1$ and $b > 1$, the number $ab-a-b+1$ is also a divisor of $n$. ...
5
votes
1answer
50 views

Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
2
votes
0answers
27 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
votes
1answer
34 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 ...
1
vote
0answers
26 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
votes
0answers
40 views

Hypothetical proof of Goldbach's conjecture? [on hold]

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
votes
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
votes
0answers
32 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 ...
0
votes
1answer
27 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, ...
0
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0answers
20 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
votes
1answer
47 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 ...
4
votes
2answers
64 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?
0
votes
0answers
53 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
votes
1answer
13 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
21 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
votes
0answers
31 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
votes
1answer
39 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 ...
8
votes
0answers
27 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 ...
0
votes
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
63 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
votes
0answers
24 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 ...
2
votes
2answers
36 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
votes
2answers
51 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 $\alpha, p \in \mathbb{Z}$ ...
-3
votes
1answer
42 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
91 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
votes
1answer
36 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
vote
1answer
47 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
votes
1answer
11 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
vote
3answers
59 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
44 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
votes
0answers
39 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
vote
1answer
65 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
votes
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
87 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
37 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
votes
0answers
62 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 ...