3
votes
1answer
48 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
54 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 ...
0
votes
0answers
61 views

Is my proof good enough?

Prove: The product of any three consecutive natural numbers is divisible by 6. 6|n (n+1) (n+2). If n is an odd number (n=(2 t+1), t is any natural number), then (n+1)= ((2 t+1)+1) an even number. So ...
3
votes
2answers
42 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
60 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
74 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
49 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
31 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
32 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 ...
17
votes
2answers
661 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 ...
2
votes
3answers
65 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
70 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
50 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
49 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
55 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
60 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
61 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
156 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
92 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
43 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
51 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 ...
18
votes
1answer
405 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
134 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
533 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
279 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
48 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
201 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
59 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
192 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
61 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
211 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
140 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
200 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
174 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
66 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
465 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
158 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
100 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 ...
4
votes
2answers
169 views

Does this equation have integer solutions

Let $g\geq 2$ be an integer. (It will be the genus of some curve.) Are there positive integers $d$ and $e$ such that the equality $$ (e-2)(e-1) = 2d(g-1)+2$$ holds?
2
votes
1answer
269 views

Class number/ quadratic field/divisibility

If we take $n$ an even integer and greater than $5$, then $\mathbb Q(\sqrt{1 - 4k^n})$ are divisible by $n$, other than for $k = 13$ and $n = 8$. Why this is happened? If we take $n$ less than 5 (I ...
5
votes
1answer
390 views

Prove that the equation $y^2=x^3-73$ has no integer solutions

Prove that there are no integers $x,y$ such that $y^2=x^3-73$. Thank you.
0
votes
1answer
134 views

Ideals as a product of prime ideals

Suppose we are working in $\Bbb{Q}(\sqrt{-41})$. Given a ideal, for example $(2-\sqrt{-41})$ (we especially work in $\Bbb{Z}[\omega_{-41}]$). We know that this is a Dedekind ring, thus we have unique ...
3
votes
2answers
171 views

Set of sum of cosines is finite

Let $x,y \in \mathbb{R}$ such that the set $\{\cos{(n\pi x)} + \cos{(n\pi y)} | n \in \mathbb{N} \}$ is finite. Show that $x$ and $y$ are rational. I have been trying to consider a graph of this set ...
4
votes
2answers
119 views

Find the number of integral solutions of $(x,y)$

Given this equation: $4x^3+5=y^2$ Find the ordered pairs of $(x,y)$ where $x,y\in Z$
3
votes
1answer
281 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
0
votes
2answers
124 views

The discriminant of an integral binary quadratic form and the discriminant of a quadratic number field

Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversely ...
2
votes
0answers
136 views

A characterization of an ambiguous class of binary quadratic forms of discriminant $D$

We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...