0
votes
1answer
42 views

Summation and product over $k$ with $k$ prime to $n$ sought

I just come to a standstill with the following two formulas. If $$E_n=\lbrace k\mid 1\le k\le n\ \&\ (k,n)=1\rbrace$$ then I hope for a closed formula $f(n)$ for those $$\sum_{E_n}k$$ ...
1
vote
2answers
52 views

Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
5
votes
1answer
85 views

Irrational to power of itself is natural

I've been thinking about a natural number like $n$ so that $x^x=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some ...
0
votes
1answer
41 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
9
votes
5answers
215 views

Are numbers of the form $n^2+n+17$ always prime

Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it ...
6
votes
1answer
124 views

What is the largest possible length of a prime number?

Let $p$ be a prime number , set $f(p)=2p+1$ and define $f^n(p)=f\circ f\circ\cdots\circ f(p)$ composition by $f,$ $n$ times. And define length of $p$, $L(p)$ as maximum of $n$ such that $f^i(p)$ is ...
3
votes
2answers
71 views

Number of algebraic integer divisors of an algebraic integer

Let $\alpha$ be an algebraic integer of degree $d$. Let $\tau(\alpha)$ be the number algebraic integers $\beta$ of degree $d$ such that $\alpha/\beta \in \mathbb{Z}$. What is a good upper bound on ...
0
votes
2answers
33 views

Show that $x$ is an algebraic number? Where $x$ is…

Can someone help me with the following problem? Show that $x=\sqrt2+\sqrt[3]3$ is an algebraic number. By finding a polynomial with rational coefficients for which $x$ is a root of. Can someone ...
1
vote
1answer
54 views

What is known about the ramification index of ramified primes in an arbitrary cyclotomic extension of $\mathbb{Q}$

Let $\zeta$ be a primitive $m$th root of unity, and $L = \mathbb{Q}(\zeta)$. Then $B = \mathbb{Z}[\zeta]$ is the integral closure of $\mathbb{Z}$ in $L$. If $P$ is a prime ideal of $B$ and ...
4
votes
2answers
69 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
2
votes
1answer
135 views

Quadratic reciprocity problem

How can I use quadratic reciprocity to prove that $-3$ is a quadratic residue $\pmod p$ if and only if $p=2$ or $p \equiv 1 \pmod 6$ and deduce that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_p ...
2
votes
1answer
61 views

Lower bound on divisors of $\Phi_n(n) $

Take the nth cyclotomic polynomial $\Phi_n(x)$ and let $\phi$ be the Euler totient function. I can prove that all divisors $d$ of $\Phi_n(n)$ are such that $d \ge \phi(n)$ or $d = 1$. The proof is ...
3
votes
2answers
46 views

Number theory problem - contradiction

In an algebraic proof (for my problem it doesn't matter which proof) I have a special setting: $a,b,c \in \mathbb{Z}, \text{gcd}(a,c)=1,b<c \ \text{and} \ a \in \left\lbrace 1, \ldots , ...
1
vote
0answers
70 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
0
votes
4answers
83 views

Ordering the solutions to Pell's Equation

Let $S$ be the set of positive integer pairs $(x,y)$ such that $x^2 - d y^2 = -4$ or $x^2 - d y^2 = 4$, where $d$ is fixed as the discriminant of a real quadratic number field. I'm trying to show ...
2
votes
1answer
60 views

Sum of Three squares

Considering $z=a+bi+cj$ ($a,b,c\in\mathbb{Z}$) and $w=d+ei+fj$ ($d,e,f\in\mathbb{Z}$) and the property of complex numbers that $|zw|=|z||w|$. If the rule of multiplication $zw$ is defined such that ...
0
votes
1answer
34 views

$p$ is an odd prime of the form $p=x^2+2y^2$ iff $p\equiv_8$ $1$ or $3$ [duplicate]

How would I prove the following: Show that an odd prime $p$ can be written on the form $p=x^2+2y^2$ for some $x,y\in\mathbb Z$ iff $p\equiv_8 1, 3$. Hint: use the quadratic reciprocity and the ...
1
vote
1answer
41 views

Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
16
votes
2answers
796 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
1
vote
3answers
96 views

Legendre symbol question for infinitely many primes of form 4k + 3

Given a positive integer n, how would one show that there are infinitely many primes p of the form 4k + 3 that have Legendre symbol (n/p) = -1? From the comments I have received thus far, it has been ...
1
vote
3answers
99 views

Divisors of $mn$ in Rings without Unique Prime Factorization

Using the fundamental theorem of arithmetic, it's easy to prove this proposition: Proposition. Every divisor of $mn$ can be written as the product of a divisor of $m$ to a divisor of $n$. My ...
0
votes
1answer
67 views

How to calculate such sums of Legendre symbols?

How to calculate such sums as $\sum_{x\in\mathbb{F}_p} \left(\frac{x^2+ax+b}{p} \right)$ If $x^2+ax+b$has a root, $b$ may be eliminated and the sum is evaluated to be $0+\sum_{x\in\mathbb{F}_p^*} ...
0
votes
1answer
52 views

set theory, show sets are not of equal cardinality - check my proof

question from exam in set theory: let $M$ be the set of all real numbers x that satisfy: $cx^2+bx+a=0$ where $a,b,c \in Z$ (Meaning they are integers) and $c$ is not $0$. We will define $K = \{sm+t ...
0
votes
1answer
57 views

Kernel of a homomorphism is subgroup of squares

Let $\gamma:(\mathbb{Z}/p^m\mathbb{Z})^*\rightarrow \{1,-1\}$ be defined as $\gamma(a)=(\frac{a}{p})$, the Legendre symbol; $p$ is an odd prime, and $m$ is an integer greater or equal to $1$. I have ...
4
votes
1answer
65 views

perfect squares possible?

If we let a, b, c, d, and x be integers is it possible that $$x^2+a^2 = (x+1)^2 + b^2 = (x+2)^2 + c^2 = (x+3)^2 + d^2$$ My initial thought is no way! I tried expanding and simplifying, getting $$a^2 ...
0
votes
0answers
24 views

How is this defined for m when $\gcd(m,D)>1$?

If $D\equiv 0,1 \pmod{4}$ is a nonzero integer, then there is a unique homomorphism $\chi:(\mathbb{Z}/D\mathbb{Z})^*\to \{1,-1\}$ such that $\chi([p])=\bigl(\frac{D}{p}\bigr)$ for odd primes $p$ not ...
2
votes
0answers
70 views

Let n be a nonzero integer and p an odd prime not dividing n.

$p/(x^2+ny^2)$ for $x$, $y$ relatively prime $\Leftrightarrow$ to $(\frac{-n}{p})=1$. I have proved the "$\Rightarrow$" part by using the fact that $y$ and $p$ must be relatively prime, which implies ...
0
votes
1answer
222 views

Why perfect square has odd number of factors

can someone please describe me why only the perfect square has odd number of factors.why does other number not has odd numbers of factors? I understand it but don't find any mathmetical proof.Please ...
1
vote
0answers
103 views

$ p = a^{2} + ab +b^{2} \ a, b \in \mathbb{Z} $

Let $p \neq 3 $ be a prime. Prove that $ p = a^{2} + ab +b^{2} \ a, b \in \mathbb{Z} \iff p \equiv 1 \ mod \ 3$. The $\rightarrow $ direction is easy. For the other implication, I considered ...
0
votes
0answers
44 views

$p=a^2-3b^2$ or $p=3b^2-a^2$, $p$ prime

Consider the number field $K=\mathbb{Q}[\sqrt{3}]$, with number ring $\mathbb{Z}[\sqrt{3}]$. Now, $3$ is a square modulo a prime $p$ iff the minimal polynomial $x^2-3$ of $\sqrt{3}$ splits in ...
0
votes
0answers
57 views

Quadratic Diophantine Primality Testing

Define a 2-Quadratic Group Operation as the following: A 2nd degree polynomial of the form: $$a_1x_1 + a_2x_2 + a_3x_1^2 + a_4x_2^2 + a_5x_1x_2 $$ Define a primal 2-quadratic group number as an ...
19
votes
1answer
459 views

Why did Gauss think the reciprocity law so important in number theory?

Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic ...
4
votes
1answer
154 views

Prerequisites for Dirichlet & Dedekind's Vorlesungen über Zahlentheorie

What are the prerequisites for Dirichlet & Dedekind's Vorlesungen über Zahlentheorie? There is an English translation of this book: P. G. Lejeune Dirichlet, R. Dedekind translated by John ...
0
votes
0answers
136 views

How to prove that homometric sets lead to same result in this problem? (any justifications?)

First let me define Difference multiset for a set of integers $$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j $$ as below: $$ D = \{p_i-p_j \mod N ,\quad i \ne j\} $$ I ...
9
votes
1answer
692 views

Special cases of the Stark-Heegner theorem with simple proofs

The Stark-Heegner theorem states that the ring of integers of the quadratic number field $\mathbb Q(\sqrt{m})$, where $m$ is a squarefree negative integer, is a principal ideal domain, iff ...
10
votes
2answers
395 views

Factoring a number of complex integers?

Say you are given a number (ex: $377$) and you express it in a form that allows you to factor it over the complex integers: Notice, $377 = 16^2 + 11^2$ Thus: $(16 + 11i) $ and $(16 - 11i)$ Are ...
3
votes
1answer
49 views

Express in terms of familiar arithmetic functions

How can I express the sumation $$h_k(n)=\sum_{d|n, k|d}\mu (d)$$ in terms of familiar arithmetic functions, where $k\in \mathbb{N} $ is fixed?
7
votes
2answers
208 views

Solve: $x^2-py^2=q$

Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers. It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol. ...
4
votes
1answer
61 views

On Selmer's curve

I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
4
votes
3answers
141 views

Show that there exists $f ∈ \mathbb{Z}$ such that $f^2 + f +1 ≡ 0 \pmod p$.

Let $p ≡ 1 \pmod 3$ be a prime. Show that there exists $f \in \mathbb{Z}$ such that $f^2 + f +1 \equiv 0 \pmod p$. I know the first few primes of this form are: $7,13,19$ So for example $p=7$ we ...
1
vote
1answer
199 views

For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically? kind thanks
0
votes
1answer
63 views

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
0
votes
0answers
237 views

Show that the field of p-adic numbers is complete

this is a question from a book I'm struggling with, please could you provide a clear proof Show that the field of p-adic numbers is complete i.e. that a sequence of p-adic numbers converges if and ...
4
votes
3answers
154 views

Prove or disprove that $\phi(a^n - 1)$ is divisible by n

I have a proof for the case of $a$ being prime I believe, I think this is also true for $a$ composite since I ran a test for the first $100$ numbers over the first $100$ values of $n$ and it seems to ...
0
votes
0answers
245 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
5
votes
1answer
181 views

A Gauss sum like summation

I would like to calculate the following sum. Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime. The sum is $$\sum_{j=1}^n (-1)^j ...
2
votes
0answers
71 views

How accurately can a irrational algebraic number represent a different irrational number compared to a rational approximation?

If I am trying to approximate $x = \sqrt D$ such that D is a square free integer I can use Diophantine approximation and the Fundamental Recurrence Formulas to find a rational approximation ...
15
votes
2answers
479 views

Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to ...
5
votes
1answer
165 views

Typo in Marcus' $\textit{Number Fields}$?

I am doing Problem 5.10 of Marcus where it is given that $m$ is a square-free negative integer and that $\mathcal{O}_K$ is a PID where $K = \Bbb{Q}(\sqrt{m})$. Now in part (b) of this problem he ...
5
votes
1answer
107 views

Quadratic Diophantic equation

Hello :) i want to give a answer op the following question: For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$. I want to solve the question without ...