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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Lagrange Multipliers for farthest distance

I am trying to find the farthest point from the origin to a point on the circle $$(x-2)^2+y^2=1$$ I am not great with the formatting on here but this is what I have so far... $$f(x,y)=x^2+y^2 $$ ...
4
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0answers
30 views

Proving that the number of integer solutions of $x^2-Ny^2=1$ is infinite

I am trying to prove that the number of integer solutions of $x^2-Ny^2=1$ is infinite whenever N is a squarefree integer. For this I define norm of $a+b\sqrt N=a^2-Nb^2$. Now I prove that $a+b \sqrt ...
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0answers
18 views

Proving that a summation is multiplicative

I have been give a project for number theory: For m>0 , let f(m) = $\sum_{r=1}^{m} \frac{m}{gcd(m,r)}$ . Evaluate f(m) in terms of the prime factorization of m. So far, I have found a formula for ...
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0answers
14 views

explicitly splitting the hamilton quaternions over local fields

For simplicity, lets first consider the hamilton quaternions $$ H = \left(\frac{-1,-1}{\mathbb{Q}}\right)$$ This is the central division algebra over $\mathbb{Q}$ with $\mathbb{Q}$-basis given by ...
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0answers
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$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
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1answer
29 views

Power to which 7 is raised to produce a number starting with 2015…

Q. Is there a power of 7 such that the number produced starts with 2015? I am completely stumped and any hints towards the solution would be great, bear in mind this is a question posed to a student ...
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2answers
15 views

Can some of the case of this congruence be solvable? And what is the general way to solve this if it is solvable?

$a^m$ congruence to 1 (mod n) where a and n is not a coprime and m is an integer. How do you prove it if it is not solvable?
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0answers
7 views

How does class field theory help us deduce the splitting of nonprincipal prime ideals?

I had a general question about the significance of global class field theory. One of the goals, as I understand, is to answer the following question: Given $L/K$ abelian, $g$ a divisor of $[L : ...
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2answers
16 views

If every prime that divides $n$ also divides $m$, show that $\phi(mn)=n\phi(m)$ & $\phi(mn)=m\phi(n)$

If every prime that divides $n$ also divides $m$, show that $\phi(mn)=n\phi(m)$ & $\phi(mn)=m\phi(n)$ My attempt. As every prime that divides $n$ also divides $m$ implies $(m,n)=d$ where ...
2
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1answer
31 views

Largest possible subset primes

Let $q$ be a Sophie Germain prime number, i.e. $2q+1=p$ is prime. Consider the set $\{1,2,3,\ldots,p-1\}$. Then what is the maximum size of a subset of this set, such that the subset contains no two ...
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0answers
23 views

Integers points of an elliptic curve

I am concerned by the number N of integer points in some class of elliptic curves. It is known to be finite for each elliptic curve C the corresponding bound being a function $N_C$ which gives a huge ...
2
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1answer
11 views

number of weak compositions modulo prime $p$

For $n\in \mathbb{N}$ and some prime $p$, consider $(\mathbb{F}_p)^n$. Is it known how many weak compositions $$x_1+x_2+\ldots +x_n\equiv 0 \pmod p$$ in $\mathbb{F}_p$ there are, where $(x_1, \ldots, ...
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0answers
23 views

Quadratic Field of $Q[√−1]$…

Can someone show me a complex plane around the origin, with the points on the part of the complex plane which are quadratic integers in $Q[√−1]$. Another graph for $Q[√−3]$. And another for $Q[√−5]$. ...
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1answer
41 views

Is there a fast divisibility check for a fixed divisor?

Is there a fast algorithm to check if $d \mid n$ is true for varying $n$, if divisor $d$ is fixed? Variable $n$ is a $w$-bit binary integer, $d$ is an integer constant.
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3answers
37 views

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $n$

Prove that $\phi(n)=\frac{n}{2}$ iff $n=2^k$ for some integer $n$ Attempt: Let $n=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k}$. Then $\phi(n)=\frac{n}{2} \implies ...
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1answer
9 views

What other types of distributivity are there?

When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like. For example, to distribute $x$ into $\text{id}_y$, you simply have ...
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2answers
45 views

A bizarre property

Studing some fact about p-adic numbers I read a bizarre property. A metric space S is called ultrametric when $d(x, y) \le\max\{d(x, z), d(z, y)\} \forall(x,y,z) \in S^3$. Prove that all ball of S ...
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1answer
17 views

Primes in Quadratic Fields with Norm less than 6

What are the primes in $\mathbb Q[\sqrt{−1}]$ which have norm less than $6$? Also what primes in $\mathbb Q[\sqrt{−3}]$ have norm less than $6$, and the primes in $\mathbb Q[\sqrt{−5}]$? Which of them ...
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0answers
36 views

Class Group of $\mathbb Q(\sqrt{-15})$

Class Group of $\mathbb Q(\sqrt{-15})$ I used this paper for my attempt. First the discriminant of $\mathbb Q(\sqrt{-15})$ is the discriminant of the monic minimal polynomial of ...
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2answers
37 views

What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument?

How is called the subset of Gaussian integers such that from all Gaussian integers having the same argument only one with the smallest absolute value is included? Is there a special name for them? ...
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1answer
21 views

Show that $ζ$ is a Quadratic Integer in $Q[\sqrt{−3}]$

So in the complex plane, there are three cube roots of one. Suppose we let $ζ$ be the cube root of one which has positive imaginary part. How can we show that $ζ$ is a quadratic integer in ...
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1answer
51 views

How Deficient a Number is? (Finding numbers having a certain deficiency)

This question was edited, in particular equations were corrected: A number N is said to be deficient by an integer $d$ if: $\sigma(N)=2N-d$ Note that powers of 2 are deficient by 1. While a prime ...
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2answers
30 views

Purely number theory problems

Suppose the numbers $1,2,3,\dots,1986$ in any order are concatenated then prove that the number is not a perfect cube. This problem gives me a feeling that here cubic residues can only help no other ...
2
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1answer
31 views

Average difference between two odd numbers of equal length

If I select two different odd numbers of binary length $l$, what is the formula that will tell me the average difference between those two numbers? Note that the high order digit must always be $1$, ...
11
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1answer
152 views

A proof involving the Euler phi function

Problem: Let $\varphi$ be the Euler phi function, where for any $n \in \mathbb{Z^+}$, $\varphi(n)$ is the number of positive integers less than $n$ that are relatively prime with $n$. ...
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1answer
37 views

What is the number of complex integers inside a circle of radius r? [on hold]

What is the number of such complex integers, $z$, that $|z|\le r$? I am interested in a closed-form formula for integer $r$.
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2answers
47 views

Solving $a^2$ $+$ $ b^2$ $=$ $2c^2$ [on hold]

I was working through some number theory problems , when I came across the following question : Find all solutions of $a^2$ $+$ $b^2$ $=$ $2c^2$ Can someone help me out ? Maybe a hint ...
9
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4answers
283 views

Understanding the trivial primality test

I'm reading an algorithms book and I came across a code example for a primality test. The problem is that I couldn't understand the condition for the for-loop: ...
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3answers
97 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
3
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3answers
30 views

Largest subset with no arithmetic progression

I am trying to find some weak bounds on the largest subset of a set, such that the subset has the property that it contains no three elements in arithmetic progression. The elements of the original ...
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3answers
55 views

For any 7 different real numbers, there are among them two numbers x and y such that $0<\frac{x-y}{1+xy} < √3$

For any 7 different real numbers, there are among them two numbers x and y such that $\frac{x-y}{1+xy}$ is greater than zeron and less than the square root of 3. I find this fact quite amazing for ...
4
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4answers
118 views

How to solve $x^3\equiv 10 \pmod{990}$? [on hold]

How to solve $x^3\equiv 10\pmod{990}$? It has 3 solutions: 10, 340, 670. Here is the link: https://www.wolframalpha.com/input/?i=x%5E3+%3D+10+%28mod+990%29
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1answer
67 views

Prove that there are infinitely many primes in $\mathbf Q[\sqrt{d}]$.

Prove that there are infinitely many primes in $\mathbb Z[\sqrt{d}]$. I don't know how to prove this, but I think that the proof will be similar to proving that there are infinitely many primes in ...
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1answer
64 views

Is there an arbitray large prime equal to 3k+2? [duplicate]

How could I find an arbitray large prime number equal to 3k+2?
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1answer
36 views

If two integers are associates then their norms are equal in absolute value [on hold]

In $Q[\sqrt{d}]$ prove that if two integers are associates then their norms are equal in absolute value when $d>0$ and $d<0$
2
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1answer
17 views

Infiniteness of set of primes such $f$ have root $\mod p$ [duplicate]

Let $f \in \mathbb{Z}[x]$ be non constant. How to prove that exists infinitely many primes such $f$ have root in $\mathbb{Z/_{(p)}}$. I spent much time, but with no benefits.
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0answers
22 views

define a notion of congruence [on hold]

So I am learning congruences and ring theory etc, and I have a question. If $α$ is a quadratic integer in $Q[√−d]$, then what would define a notion of congruence (meaning mod $α$). Also, how would ...
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2answers
41 views

Sum of odd Fibonacci Numbers

Trying to prove that the sum of odd-index consecutive Fibonacci numbers is the next even-index Fibonacci number. I have a gap in my proof that I cannot figure out. I know that induction would be ...
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0answers
48 views

Choosing M cards from N decks

Alice and Bob are playing cards. They have N decks of cards. Each deck of cards contain 52 cards: ...
2
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0answers
10 views

norm map and local class field theory

Let $K$ be a local field, say a finite extension of $\mathbb{Q}_p$ (which is the purpose of my interest). Let $L$ be an unramified extension of $K$. Local class field theory asserts that there ...
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2answers
24 views

Congruence Classes $(\text{mod} (3 + \sqrt{−3})/2)$ in $Q[\sqrt{−3}]$ [on hold]

What would be the congruence classes $(\text{mod} (3 + \sqrt{−3})/2)$ in $Q[\sqrt{−3}]$?
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votes
3answers
40 views

Number of times $2^k$ appears in factorial

For what $n$ does: $2^n | 19!18!...1!$? I checked how many times $2^1$ appears: It appears in, $2!, 3!, 4!... 19!$ meaning, $2^{18}$ I checked how many times $2^2 = 4$ appears: It appears in, ...
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1answer
48 views

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors …

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors $a$ and $b$ of $n$, the number $a + b − 1$ is also a divisor of $n$. This was taken from the Russian ...
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3answers
41 views

How to apply Chinese Remainder Theorem for $x$

If: $$x \equiv 0 \pmod{17}$$ and $$x \equiv -1 \pmod{9}$$ Then how is: $$x \equiv 17 \pmod{153}$$ I get that since $\gcd(9, 17) = 153 $ the solution will be $\pmod{153}$ but how do you get the $17 ...
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1answer
11 views

Ideal factorization Theorem, more generally

Consider Theorem 4.3.1 in link (it's quite long, so please open the pdf) I'm wondering if we can assume that the prime ideal we want to decompose is not $(p)$ with $p$ a prime in $\mathbb Q$, but a ...
5
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5answers
74 views

solutions such that a combination number is odd

Let $m$ be a positive integer. Given $m$, I want to find the largest $n$, $1\leq n\leq m$, such that $$m+n\choose n $$ is odd. Is there any standard theorems or results? Any references? Thanks!
1
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1answer
23 views

Number of $q$-th residues modulo $n$

Let $q$ be a prime and $n\ge 2$ an integer. Moreover, define $f_q(n)$ as the number of $q$-th residues modulo $n$. Is it true that if $K$ is a positive constant then there exist infinitely many $n$ ...
0
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0answers
16 views

Jacobians and ranks of a curve

I would like to know the following: How to find Jacobian and rank of an hyper elliptic curve like $x^5-x= y^2-y$? High regards Rosy
11
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0answers
29 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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1answer
27 views

Prove that there is an integer a such that a is a primitive root modulo p^2 and a is relatively prime to n. [Hint: Use the Chinese Remainder Theorem.] [on hold]

let n be a natural number, let p be a prime, and suppose $p^2 \mid n$. Prove that there is an integer a such that a is a primitive root modulo $p^2$ and a is relatively prime to n. [Hint: Use the ...