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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Prove that there is no integer $k$ with $P(k)=8$

Let $P(x)= x^n + a_{n-1}x^{n-1}+...+a_1x+a_0$be a polynomial with integral coefficients. Suppose that there exists four distinct integers $a$, $b$, $c$, $d$ with $P(a)=P(b)=P(c)=P(d)=5$. Prove ...
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48 views

Composite expressions of the form $a^n +1$

I call a diophantine expression $f(n)$ a composite expression if $f(n)$ is a composite number for every integer $n$. I am interested in expressions of the form $a^n + 1$. It is trivially seen that ...
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52 views

Triples of positive integers

Prove that there is only a finite number of triples of $a,b,c \in \mathbb{Z}$ such that $\gcd(a,b) = c$, $\text{lcm}(a,c)=b$, and $a^n + b^n = c^n$ for some $n \in \mathbb{Z}^+$. Thank you for the ...
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19 views

Can sum of first $n$ primes be represented as a weighted sum of first $k<n$ primes with weights sum to $n$?

For the sum of first $n$ primes, denoted as $x_n = \sum_{i=1}^n p_i$ with $p_i$ representing $i$th prime, can $x_n$ be represented as a weighted sum: $x_n = \sum_{i=1}^k c_i p_i$ with $k<n$ and ...
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41 views

why this $ab+k,bc+k,ca+k$ always perfect square

let prime number $p$ such $p\equiv 1\pmod{12}$,and $p|k$,show that:there exist infinitely many triples $(a,b,c)$ of postive integers such that $a,b,c$ are in arithmetic progression,and ...
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35 views

Let $p$, $q$ be prime numbers, $p \neq q$. Find a natural number $n$ with $0 \neq n < pq$ such that $p^{2q−1} + q^{2p−1} \equiv n \pmod {pq}$.

Let $p$, $q$ be prime numbers, $p \neq q$. Find a natural number $n$ with $0 \neq n < pq$ such that $p^{2q−1} + q^{2p−1} ≡ n $(mod $pq)$. So I figured out that $p^{2q−1} = p$ mod $q$ and $q^{2p−1} ...
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52 views

Number Theory: Problems with Integer Squares

I have these problems assigned for homework, I was able to get through the rest of the homework fine but I'm stuck on these two. Prove that If $n\equiv 6\pmod9$, then $n$ cannot be a sum of two ...
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41 views

What is the best approximation of the 'prime-counting function'?

I found that the 'prime-couting function' is given by this formula: $$\pi(x)=\sum_{k=2}^{\left\lfloor x\right\rfloor} \left\lfloor\frac{\phi(k)}{k-1}\right\rfloor$$ With $\phi(k)$ is the Euler's ...
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29 views

are the subgroups of $(\mathbb{Z}/20\mathbb{Z})^{\times}$ cyclic?

I found 8 subgroups of $(\mathbb{Z}/20\mathbb{Z})^{\times}$, namely the subgroups generated by each one of its elements. Are there any others? If so, how am I supposed to find all of those that are ...
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36 views

An application of congruences: cards shuffle

Consider a deck of cards. Take the bottom and top cards of the deck and place them on the table to start a new deck. Then take the remaining bottom and top cards and place them on the newly started ...
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46 views

How small can $\frac{p_n}{p_1}$ become, if $\prod_{j=1}^n p_j$ is a Carmichael-number?

Suppose, $N=\prod_{j=1}^n p_j$ is a Carmichael-number. Define $c(N)=\frac{p_n}{p_1}$. The numbers $p_1,...,p_n$ are the prime factors of the carmichael-number $N$. $c(N)$ is the ratio of the largest ...
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23 views

What is Mordell's conjecture in simple terms?

I am reading Ian Stewart's The Great Mathematical Problems, and come across the chapter regarding the Mordell's conjecture. After reading I couldn't draw a clear picture about the conjecture and also ...
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66 views

Estimates of the sum involving both the Mobius function and Mertens function.

I want to ask on the estimates of the sum $$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$ But it is little known ...
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43 views

Can every odd prime $p\ne 11$ be the smallest prime factor of a carmichael-number with $3$ prime factors?

According to my search, the number $561=3\times11\times17$ is the only carmichael-number with $3$ prime factors, which is divisible by $11$. Is this true ? If yes, $11$ cannot be the smallest ...
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26 views

Primes in integral closure of Gaussian integers field

Let us have $A[\sqrt{13}] = \mathbb Z[(1+\sqrt{13})/2]$. How can all the prime elements of it be described? $z \in Z$ is prime if quotient ring $Z[\Theta]/(z)_{Z[\Theta]}$ is a field. I have ...
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23 views

Can the bounds for the number of carmichael numbers below $x$ be made more concrete?

Here https://en.wikipedia.org/wiki/Carmichael_number at the part distribution lower and upper bounds for the number of carmichael numbers below $x$ (denoted by $C(x)$) are given. Two questions ...
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42 views

Are there non-abelian totally real extensions?

I'm aware that abelian extensions are either totally real or CM. Also, Galois extension are either totally real o totally imaginary. But I'm wondering about the converses to those statements. Are ...
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94 views

When does linear combination of real-rooted entire functions of genus 0 or 1 remain real-rooted?

In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions: $$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$ where ...
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39 views

Conjugacy Class in Galois Representations

Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$ \rho: G ...
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Is the following proof correct of why $\gcd(a,b)$ smallest linear combination of $a$ and $b$?

This is the proof I have: Lets see why $\gcd(a, b) $ is the smallest positive linear combination of $a$ and $b$: Let $LC = \{ s'a + t'b : s', t' \in \mathbb{Z}, s'a + t'b > 0 \}$. By the ...
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34 views

Periods of Fibonacci numbers in mod. Number Theory

Work out the periods $π(n)$ of the $\mod n$ such that $$f_k ≡ f_{k+π(n)} \mod n$$ I got $π(2)=3$,$π(3)=8$,$π(4)=6$ by computing it and looking at the periods. Now Part 2 Prove that for all ...
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36 views

Rational points of conics over $\mathbb{Q}$

I am starting to read lecture notes on basics of arithmetic geometry by A. V. Sutherland. In the second lecture, there is a procedure how to decide whether a conic over $\mathbb{Q}$ has a rational ...
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31 views

Invariance of a set under permutations

Given a positive integer $n$, let $f$ be a function $$ \{1,\ldots,n\}^2 \to \{1,\ldots,n\}. $$ Then, it is possible that there exists a permutation $\{\sigma_1,\ldots,\sigma_n\}$ of $\{1,\ldots,n\}$ ...
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45 views

solving for $P=a^{2}+b^{2}$

$ Fermat's Two-Square Theorem:$ Given a prime $ p$, there exist integers $ a, b$ such that $ a^2 + b^2 = p$ iff $ p = 2$ or $ p \equiv 1 \bmod 4$. Consequently, a number $ n$ is expressible in the ...
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51 views

Does $\mathbb{Z}/p^2\mathbb{Z}[x]$ have unique factorisation?

Let $p$ be a prime greater than $2$. Does $$\left(\mathbb{Z}/p^2\mathbb{Z}\right)[x]$$ have unique factorisation ? Of course this is valid if one considers ...
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37 views

modern proof of the conditional three prime theorem by Hardy and Littlewood

Hardy and Littlewood proved the three prime theorem under the GRH(generalized Riemann hypothesis) in an old paper: Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of ...
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41 views

Positive integers $\gt 5$ that cannot be expressed as the sum of a semiprime and a prime

Let $n \in \mathbb{N}$ and $n \gt 5$ so that $$n \neq \text{prime} \times \text{prime} + \text{prime}$$ So far I only found $n = 10$. Are there more?
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21 views

Selberg's sieve: how to evaluate the main part of the asymptotic expansion?

I'm currently reading the presentation of Selberg's sieve by Gelfond & Linnik, Elementary methods in the elementary theory of numbers. I have difficulties in evaluating the sum $$\sum_{d \leq ...
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40 views

How to derive a formula related to the Gauss sum

Let $\chi$ be a Dirichlet character modulo $m$ induced by $\chi'$ modulo $m'$. We define $$ \tau(\chi):= \sum_{a(mod \ m)} \chi(a) e(a/m). $$ Could someone please show me how to derive the formula: ...
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45 views

How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All: I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com: ...
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28 views

Can the Von-Mangoldt function and the Chebyshev function be defined for entire complex plane?

Can the von-Mangoldt function and the Chebyshev function be defined for the entire complex plane ? I assume so, but I had not seen the definition. Can anyone provide some links for this? Thank you.
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41 views

for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$)

Prove that for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$) I don't think this statement is true since $k$ the sequence $k \cdot ...
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26 views

Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on ...
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39 views

Finding numbers of order 10 in various groups help.

So I actually have a solution, but I need help understanding it. Here is the problem. I don't understand why the numbers relatively prime to 10 are of order 10. And also why there are none in ...
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42 views

How can I create new factorization projects?

I found many factorization projects in the internet : Mersenne-primes, Fermat-primes, Cunningham-project , Proth-number-search and many more. If I want to create a new fatorization project : 1) ...
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65 views

Problems from Guy's Unsolved in Number Theory

Are the problems from Richard Guy's book ''Unsolved Problems in Number Theory'' really unsolved? Is it possible some of those problems tobe the main topic of someone's PhD?
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64 views

Results regarding the relative-totient function

Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function. For now assume $x>n$. Is there a general formula for ...
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48 views

Elementary theorems in number theory which need analytical methods (analysis)

A well known theorem of Dirichlet says: if $a,b$ are relatively prime positive integers, then there are infinitely many primes of the form $an+b$. The original proof by Dirichlet (possibly) uses ...
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How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
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67 views

How to prove that $V\otimes_{\mathfrak o}\mathfrak o_{\mathfrak p} \approx V\otimes_K K_\mathfrak p$

Let $\mathfrak o$ be a Dedekind domain with field of fraction $K$, $\mathfrak p$ be one of its prime ideal, $K_{\mathfrak p}$ be the completion of $K$ at $\mathfrak p$, i.e. with respect to $| ...
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349 views

Number Theory Prove $\gcd(a, b, c)=\gcd(\gcd(a,b),c)=\gcd(\gcd(a,c),b)=\gcd(\gcd(b,c),a)$.

Let $a, b, c$ be integers, no two of which are zero, and $d=\gcd(a, b, c)$. Show that $d=\gcd(\gcd(a,b),c)=\gcd(a,\gcd(b,c))=\gcd(\gcd(a,c),b)$. Here is what I have tried, but I'm unsure if the part ...
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42 views

Why is $M(x)=\frac{1}{x}\sum_{n\leq x}\mu(n)=o(1)$ ($x\to\infty$) equivalent to the Prime Number Theorem

Where $\mu$ is the mobius function and $o(\phi)$ is a Lindau Symbol, $f=o(\phi)$ if and only if $f/\phi\to 0$. Found it stated as an obvious fact in a math journal. I'd like a hint if it really is ...
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26 views

Statement about abelian class field theory

I am reading Lectures on the langlands program...(http://arxiv.org/pdf/hep-th/0512172.pdf) by EDWARD FRENKEL He mentioned the statement of abelian class field theory as Galois group ...
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45 views

Integral representations of $\zeta(s)$ using the floor/frac functions. How could this one be derived?

Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s ...
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25 views

Proof that coprime cubes are divisible by $n$

I would like to ask if my proof is correct. Task is to prove that $$ n \big| \left( \sum_{0 < a < n \atop (n, a) = 1} a^3 \right), $$ where $n>2$. My proof: Take a number $k$ coprime to ...
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49 views

A conjecture on integers coprime to $2^n-1$ and having a prescribed Hamming weight in their binary representation

I wonder if anyone has seen this before or would have some ideas on how to go about proving it. I have done several experiments with the computer and seems to hold. For an integer $k$, denote by ...
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54 views

The number of totatives to the nth primorial, in an interval shorter than the nth primorial.

Can, and if so when can, we determine the amount of natural numbers which are relatively prime to the nth primorial in an interval of length smaller than the nth primorial? Introduction Let ...
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72 views

A different viewpoint of Hilbert's Theorem 90

Let $L/K$ be a galois extension with galois group $G$($|G| = n$) cyclic and generated by $\sigma$. Let $\beta \in L$ have $N(\beta) = 1$. $N(.)$ is the norm function from $L$ to $K$. Hilbert's ...
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48 views

Proving that for $F_k = F_{k-1} + F_{k-2}$, $F_k$ is even iff $3|k$

Consider the recursion $F_k = F_{k-1} + F_{k-2},$ $ k\geq 2,$ $F_0 = 0$ and $F_1 = 1$, then show that $F_k$ is even iff $3|k$ I tried to do a proof by induction: The statement is true for base ...
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32 views

element of an $\ell$-adic Galois representation with certain eigenvalues.

Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation $$ \varphi_{\ell}: \mathscr{G} \longrightarrow ...