Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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5
votes
3answers
245 views

Factoring numbers “of the (binary quadratic) form” in two different ways

For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$ I think that if $Q$ represents $m$ in two different ways then $m$ is composite. I can prove this for $n$ prime. I was hoping ...
26
votes
1answer
1k views

Relation between the Dedekind Zeta Function and Quadratic Reciprocity

I was trying to learn a little about the Dedekind zeta function. The first place I looked at was obviously the Wikipedia article above. So my question comes from a sentence by the end of the article ...
4
votes
1answer
279 views

Positive integer solutions of $x^2+21y^2=z^4 $

Can one find all positive integer solutions of $$x^2+21y^2=z^4 ?$$ I am not sure if this is possible. I just saw this problem and this problem came to my mind.
3
votes
1answer
87 views

Dividing the ramification index of an extension

Let $f(x)$ be a polynomial of degree $m$ over $\mathbb{Q}_{p}$ with all roots $r_{i}$ such that $\operatorname{ord}_{p} r_{i} = \frac{1}{p}$. Why does $p$ have to divide the ramification index of ...
4
votes
0answers
272 views

Prime zeta definition, multiplication by zero

Wikipedia has a page about the prime zeta function which is defined as follows: $$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$ I entered this additional definition: Define a sequence: ...
3
votes
2answers
281 views

$n = 2^k + 1$ is a prime iff $3^{\frac{n-1}{2}} \equiv -1 \pmod n$

Let $k \geq 2$ be a positive integer and let $n=2^k+1$. How can I prove that $n$ is a prime number if and only if $$3^{\frac{n-1}{2}} \equiv -1 \pmod n.$$ Fixed.
4
votes
3answers
1k views

Integral points on an elliptic curve

Let's start with an elliptic curve in the form $$E : y^2 = x^3 + Ax + B, \qquad A, B \in \mathbb{Z}.$$ I am wondering about integral points. I know that Siegel proved that $E$ has only finitely many ...
2
votes
1answer
92 views

All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
6
votes
5answers
3k views

Prove $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ if $\gcd(a,b) = 1$

I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$ $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ holds. I think the first step should look something like this: $d = \gcd(a+b, a-b) = \gcd(2a, a-b)$ ...
3
votes
2answers
116 views

The equation $F(x) \equiv 0 \pmod m$ has integer solution for x

Let $F(x)=(x^2-17)(x^2-19)(x^2-323)$ and let $m$ be a positive integer. How can one show that the equation $F(x) \equiv 0 \pmod m$ has an integer solution?
6
votes
2answers
167 views

$(7a+1)x^3+(7b+2)y^3+(7c+4)z^3+(7d+1)xyz=0$ does not have integer solutions

Let $a,b,c,d$ be integers. How can I prove that the equation $$(7a+1)x^3+(7b+2)y^3+(7c+4)z^3+(7d+1)xyz=0$$ Does not have an integer solution $(x,y,z)$ such that $\gcd(x,y,z)=1$?
1
vote
1answer
205 views

The last part of Tate'sThesis

Is there a comprehensive reference dealing with the last part(theory in the large!) of Tate's thesis? Why is the group of S units modulo the roots of unity a free abelian group of rank m?
5
votes
2answers
444 views

Infinitely many integer solutions for the equations $x^3+y^3+z^3=1$ and $x^3+y^3+z^3=2$

How do you show that the equation $x^3+y^3+z^3=1$ has infinitely many solutions in integers? How about $x^3+y^3+z^3=2$?
6
votes
2answers
239 views

On integer solutions of the equation $x^2+y^2+z^2=16(xy+yz+zx-1)$

Here is the question: Question. Show that the equation $$x^2+y^2+z^2=16(xy+yz+zx-1)$$ does no have integer solutions. I know a nice and easy (actually, an obvious) way to solve this problem. But I'm ...
3
votes
2answers
225 views

All primes $p,q,r$ such that $(p-q)^2+1=r$

How can one find all prime numbers $p,q,$ and $r$ such that $$(p-q)^2+1=r\ ?$$
8
votes
1answer
192 views

Diophantine impossibility and irrationality (or similar)

The Diophantine equation $$a^2 = 2 b^2$$ having no solutions is the same as $\sqrt{2}$ being irrational. Are there any Diophantine equations which are related to the irrationality of a number that is ...
2
votes
2answers
845 views

Congruence modulo prime power

In the book "A Classical Introduction to Modern Number Theory", I saw the following theorem (p. 43): If $p\neq 2$, and $p\nmid a$ then $p^{l-1}$ is the order of $(1+ap)$ mod $p^l.$ i.e. ...
19
votes
3answers
290 views

Probability distribution for the remainder of a fixed integer

In the "Notes" section of Modern Computer Algebra by Joachim Von Zur Gathen, there is a quick throwaway remark that says: Dirichlet also proves the fact, surprising at first sight, that for fixed ...
2
votes
1answer
933 views

Decode the following message which was sent using mod m=7081 and exponent k=1789 (RSA)

Decode the following message which was sent using mod $m = 7081$ and exponent $k=1789$ (RSA): $$ 5192 2604 4222 $$ I solved $\phi(7081)=6912$, and then solved the linear equation $1789u-6912v=1$ where ...
32
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
4
votes
1answer
336 views

Partition of equal size and equal sum

Let $p,q$ be odd primes. Is it always possible to partition $\{1,\ldots,pq\}$ into $p$ disjoint subsets $A_1,\cdots,A_q$ such that for all $i,j$ we have $|A_i|=|A_j|$ and $\sum_{x\in A_i}x=\sum_{y\in ...
1
vote
0answers
201 views

$(a^n +b^n)/((ab)^{n-1}+1)$ is a perfect $n^{th}$ power

Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer? Another question: ...
2
votes
2answers
216 views

All positive integers $n$ such that $n \mid 2^{n-1}+1$

How can one find all positive integers $n$ for which $n \mid 2^{n-1}+1$?
6
votes
1answer
329 views

On the equation $(a^2+1)(b^2+1)=c^2+1$

How do you find all positive integers $a,b,$ and $c$ such that $(a^2+1)(b^2+1)=c^2+1$?
3
votes
1answer
196 views

All positive integers $a,b$ such that $a^2+b \mid b^2+a$

How can one find all positive integers $a,b$ such that $a^2+b \mid b^2+a$?
5
votes
1answer
171 views

If $g$ is a primitive root of $p^2$ where $p$ is an odd prime, why is $g$ a primitive root of $p^k$ for any $k \geq 1$?

I saw this theorem referenced in a paper on performing shuffles in an array. There is a proof on pages 20-21 of the this link, but the proof is very terse and omits a lot of intuition. Is there an ...
0
votes
1answer
160 views

Basic Question about Iwasawa Theory

Is the statement of Iwasawa's theorem that for every number field $K$ there are $\mu$, $\lambda$ and $\nu$ such that for every $\mathbb{Z}_p$ extension $K_{\infty}$, the class number of any big enough ...
1
vote
2answers
2k views

number of ordered partitions of integer

Please, help me out How to evaluate the number of ordered partitions of the positive integer 5 Thanks!
8
votes
2answers
804 views

The modular curve X(N)

I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism from $Z/NZ \times Z/NZ$ to ...
5
votes
2answers
692 views

Euler's phi function and distinct primes

It is true that $\phi(p) = (p-1)$ only if p is a prime. I had also proven (I am not sure if this is a trivial fact or not) that $\phi(pq) = (p-1)(q-1)$ only if p and q are distinct primes. However, I ...
3
votes
0answers
289 views

Rational independence

The specific question is the following: I am given a set of $[L/2]$ numbers $$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$$ where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ ...
7
votes
1answer
236 views

What is a description for the following number theoretic object?

The title couldn't quite contain the question, so I didn't attempt to make it precise. I should note that this is the third or fourth question I've asked these past two days about problems I've been ...
7
votes
2answers
1k views

What do the $p$-adic roots of unity look like?

I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, ...
1
vote
1answer
191 views

Congruence subgroup of $SL_2(\mathbb{Z})$ definition

Congruence subgroups of $SL_2(\mathbb{Z})$ usually seem to be defined as a subgroup that contains $\Gamma(N) = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mod N \right\}$ for some ...
5
votes
3answers
434 views

All natural solutions of $2x^2-1=y^{15}$

How can I find all positive integers $x$ and $y$ such that $2x^2-1=y^{15}$? PS. See here.
40
votes
2answers
4k views

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
13
votes
1answer
430 views

Positive integers satisfying $x^{a+b} = a^b \cdot b$, how to show that $a=x$ and $b=x^x$?

Let $a,x,b$ be positive integers satisfying $x^{a+b} = a^b \cdot b$. How can I prove that $a=x$ and $b=x^x$?
3
votes
2answers
359 views

Where is the mistake in this incorrect proof in Eisenstein integers?

The Diophantine equation is $n^2 + n + 1 = m^3$ my attempt to solve it shows there is no solutions to this equation, but in fact there are four. I could not find my mistake so I hope someone could ...
14
votes
3answers
520 views

$x,y$ are integers satisfying $2x^2-1=y^{15}$, show that $5 \mid x$

Let $x, y >1$ be integers satisfying $2x^2-1=y^{15}$. How can I prove that $5 \mid x$?
8
votes
3answers
739 views

$x^3+48=y^4$ does not have integer (?) solutions

How does one find all positive integer solutions to the equation $x^3+48=y^4$?
3
votes
4answers
860 views

Simple formula for integer polynomial with $2\sin(2\pi/n)$ as a root?

Is there a simple formula an integer polynomial that $2\sin(2\pi/n)$ satisfies? For $2\cos(2\pi/n)$ the answer is relatively nice. For any given $n$, we have $2\cos(2\pi/n)= z + z^{-1}$ where $z = ...
2
votes
1answer
193 views

Diophantine equation from Carmichael: $2x^4 - 2y^4 = z^2$

I would to solve the Diophantine equation $2x^4 - 2y^4 = z^2$ by descent. This is an exercise from Carmichael Diophantine Analysis but I cannot do it. Since the LHS is even $2|z$, let $z = 2z'$ ...
1
vote
1answer
82 views

Polynomials f(x) of degree at most 5 forming a ring and field

Show that the set of all polynomials f(x) of degree at most 5 with integer coefficients is a ring. Is the set of such polynomials a field? I don't see how the ring of polynomials with degree at most ...
2
votes
1answer
210 views

Show $\Sigma_{n\leq X} 1/\phi(n) \sim \log(X)*\Sigma_{k=1}^\infty \mu(k)^2/(k*\phi(k))$

I would like to show that as X approaches infinity, $$\sum_{n\leq X} \frac{1}{\phi(n)} \sim \log(X)\cdot\sum_{k=1}^{\infty} \frac{\mu(k)^2}{k\cdot\phi(k)}.$$ I have already proven $$\sum_{n\leq X} ...
4
votes
1answer
162 views

Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
16
votes
4answers
3k views

Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?

This question relates to a discussion on another message board. Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My ...
8
votes
3answers
382 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
4
votes
1answer
217 views

Total ramification $p = \epsilon \pi^n$ implies $F = K(\pi)$ and minimal polynomial of $\pi$ Eisenstein

Before I state my question, let me give the set-up / "what I know". Let $F / K$ be an extension of number fields of degree $n$. Let $v$ be a (discrete) valuation on $K$ and let $$ \mathfrak{O}_{v} ...
1
vote
1answer
213 views

On the equation $3a^2-4b^3=7^c$

How does one find all integer solutions to the equation $3a^2-4b^3=7^c$?
4
votes
2answers
288 views

$\Sigma_{m|n} \mu(m)^2/\phi(m) = n/\phi(n)$?

I'm trying to prove $$\sum_{m|n} \mu(m)^2/\phi(m) = n/\phi(n)$$ My first realization was that $\mu(m)^2 = 1$ iff $m=1$ or $m$ is a squareless factor of $n$ and otherwise is 0. Let $\{1,m_1,m_2, ... ...