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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1answer
9 views

Ideal as kernel of a homomorphism

Consider the ring $\mathbb{Z}[i]$ of Gaussian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
1
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1answer
12 views

Definitions of valuations in terms of totally ordered group

Wikipedia gives a definition of valuations involving abelian totally ordered groups. So far I have only seen valuations taking values in the real numbers. Is there a reason for this generalization?
2
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0answers
13 views

A bound on number of elements less than $n$ of a $B_2[g]$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$. Let $S(n)$ be ...
1
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1answer
28 views

Parametrizing solutions of diophantine $8x^2 + y^2 = z^2$ gone wrong. Where's the mistake?

So I have $$ 8x^2 + y^2 = z^2. $$ Dividing both parts by $z$ yields $$ 8X^2+Y^2 = 1, $$ where $X$ and $Y$ are rational. Point $(0, -1)$ is on the ellipse, so I parametrize with $(X, tX - 1)$, where ...
0
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0answers
16 views

Proof of Chevalley–Warning theorem

How to prove Chevalley–Warning theorem (http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem) by using Fulton's trace formula (#$|X(\mathbb{F}_p)| \equiv \sum (-1)^i Tr(Frob_p|H^i(X, ...
3
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1answer
52 views

Euler's function $\phi$

Let $\phi(n) $ be Euler's Totient Function Let us consider $$ |\{ n \in \mathbb{N} : \phi (n) = 8 \} | = 5, $$ and $$ |\{ n \in \mathbb{N} : \phi (n) = 14 \} | = 0. $$ How would I go about ...
1
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1answer
32 views

For which $a>0$ does the equation $x^2+y^2+z^2=a$ have a solution in $\mathbb{Q}_2$?

We want to check for which $a>0$ we have that the equation $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$. $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$ for $x \in \mathbb{Z}_2^{\star}, y ...
5
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2answers
43 views

Determine all units in $\mathbb{Z}[\omega] := \{a+b\omega\mid a,b\in\mathbf{Z}\}$ where $\omega = \frac{-1 + i \sqrt{3}}{2}$

My attempt: $N(a + b\omega) = (a + b \omega)(a - b \omega) = a^2 + \omega^2 b^2$ I'm stuck here. Is my approach correct?
1
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1answer
25 views

Prove that if $p > 3$ is a prime then $2(p-3)! \pmod{p} =-1 \pmod{p}$.

Prove that if $p > 3$ is a prime then $2(p-3)! \pmod{p} =-1 \pmod{p}$.. I am totally lost; at first I thought this could be done by induction, but unfortunately this is not possible (at least I ...
1
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1answer
34 views

Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$

I need integer solutions of $x^2 + y^2 = z^2 + w^2$ parametrized. Can it be done? Thanks.
1
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1answer
43 views

Show that $-1=\sum_{0}^{\infty} (p-1)p^i$ in $\mathbf{Q}_p$

To show that in the field $\mathbb{Q}_p$, where $p$ is a prime, it holds that: $$-1=\sum_{0}^{\infty} (p-1)p^i$$ I did the following: It suffices to show that: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
0
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1answer
42 views

A fast method for factorizing $2^p-1$

I know that $\forall{d,n\in\mathbb{N}}:d|n\implies2^d-1|2^n-1$. Now, suppose that $n$ is prime - is there any fast algorithm for finding a divisor of $2^n-1$? By "divisor", I am referring to a ...
0
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3answers
53 views

Inductive step in proof of Freshman's Dream

I am trying to prove that for $K$ a field of characteristic $p$ prime, $q$ a power of $p$ and $x,y$ in $K$, $$(x+y)^q=(x^q + y^q).$$ I have the base case, and now I am trying to do the inductive ...
3
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1answer
76 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
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0answers
13 views

To solve the system of Diophantine equations.

I decided to compile a single task and to record such a system. $$\left\{\begin{aligned}&xt+yw=az^2\\&xw-yt=br^2\end{aligned}\right.$$ $a,b - $ integers that are the problem. It is clear ...
1
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1answer
26 views

Discrete Logarithm Problem

Question: Discrete Logarithm Problem: Let $g$ be a primitive root for $F_{p}$. Suppose that $x = a$ and $x = b$ are both integer solutions to the congruence $g^{x} \equiv h \pmod{p}$. Prove that $a ...
0
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1answer
48 views

If $a^2$ divides $b^3$, then $a$ divides $b$.

I want to prove or provide a counterexample to the following statement: $a^2|b^3 \Rightarrow a|b$. I know that $a^k|b^k \Rightarrow a|b$. My thought is that, e.g in the case of $k = 3$, where we ...
0
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1answer
35 views

$f(x)=x^2-a \in \mathbb{Z}[x]$ - show propositions

Let $f(x)=x^2-a \in \mathbb{Z}[x]$. $$p \in \mathbb{P}, p \neq 2, p^2 \nmid a$$ The equation $f(x)=0$ If $p \mid a $, the equation has no solution in $\mathbb{Q}_p$ Let $p \nmid a$. The ...
0
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1answer
32 views

Integer factorization simplification

I found a small improvement to the brute force algorithm for the Integer Factorization. Please tell me if there is a point to investigate it more or there are better similar ideas. I found that if ...
0
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1answer
29 views

Closed Form of n(mod7) [on hold]

For an integer n,what is the closed form as a function of n, if it exists, of n(mod7)={0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,...,n(mod7)}? The closed form of n(mod8) uses trigonometric ...
1
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1answer
37 views

pth root of unity in $p$-adic field

It is well known that $\mathbb{Q}_p(\mu_n)$ is a totally ramified extension of degree $(p-1)p^n$ if $\mu_n$ is a primitive $p^n$th root of unity. However how true is this statement for a finite ...
1
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1answer
37 views

Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p ...
0
votes
2answers
15 views

Finding $2m+1=2\alpha k+\alpha^2$ quickly

Given some positive integer $m$ I'm looking for all solutions $\alpha,k>0$ to $2m+1=2\alpha k+\alpha^2$ with $0<k^2<2m.$ Right now I'm finding these by looping over each of these possible $k$ ...
4
votes
1answer
36 views

There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p$ is a prime and $m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m ...
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0answers
12 views

The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$

Proposition Let $P(X,Y) \in \mathbb{Z}[X,Y]$. The following propositions are equivalent: The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$. For each $n \geq 0$ the equation $P(X,Y)$ has a ...
3
votes
2answers
83 views

Inverse limit of $\mathbb{Z}/n\mathbb{Z}$

I know that this is well-known fact that $$\lim\limits_\leftarrow\mathbb{Z}/n\mathbb{Z}=\prod\limits_p\mathbb{Z}_p,$$ however I don't know the rigorous proof of this. Can anyone give me the ...
2
votes
0answers
34 views

Short intervals with all numbers having the same number of prime factors

How to prove that for some $k, n_0$, for all $n \ge n_0$ it is never the case that all integers in $\{n, n+1, \dots, n + \lfloor (\log{n})^k \rfloor\}$ have exactly the same number of prime factors ...
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0answers
30 views

analysis/number theory study group (online) [on hold]

I plan on studying analysis from landau, rudin probably others and am looking for people (hopefully more than 1) where we could solve theorems/problems and ask each other questions. Online ...
5
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2answers
57 views

Proving that $\sin1 $(radian) is irrational without using Taylor Series Expansion.

In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational, and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but ...
2
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3answers
55 views

Counting the number of $\mathbb{F}_q$ points on a homogeneous polynomial

This is an area of number theory that I am not too familiar with and I would appreciate any assistance! Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic not 2 or 3. I have the ...
4
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0answers
32 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
1
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1answer
66 views

Intuituive reason why Fermats last theorem holds

I am unsure of whether it is normal, but to me, intuitively Fermats last theorem should not hold. If anyone intuitively believed it to be correct, why? Can someone explain so I understand ...
1
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1answer
28 views

How do we deduce that $\mathbb{Q}_p=\{ p^mu\mid u \in \mathbb{Z}_p^{\star}, m \in \mathbb{Z}\} \cup \{0\}$?

We know that each element $x$ of $\mathbb{Z} \setminus{\{0\}}$ has a unique representation of the form $x=p^m u\mid m \in \mathbb{N}_0, u \in \mathbb{Z}_p^{\star}$. $$\mathbb{Q}_p=\left \{ ...
1
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2answers
52 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...
1
vote
1answer
37 views

What is the probability that a random K-bit odd-number is prime?

Is it $e/K$? In an experiment that created 1000 random RSA-2048 key-pairs, 2000 random 1024-bit primes were created. It turned out that $727,709$ random candidates were generated, to create 2000 ...
3
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0answers
39 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
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0answers
22 views

Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin ...
2
votes
2answers
30 views

a root of some polynomial over finite field

I think this is a really basic question, but it had been a little while since I dealt with this material and I was hoping to get a bit of assistance here. Let $q = p ^{2M}$ for some prime $p$ and $M ...
0
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0answers
60 views

Idele for a rational number $q=\frac{63}{550}$ [on hold]

Wikipedia, in its article "p-adic number", has taken an arbitray number $x= \frac{63}{550}$ to show the p-adic absolute value with respect to different primes. Obviously, the p-adic absolute value is ...
0
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1answer
14 views

solve $k(k-1) \geq \ln2*2m$ for k

My Question is related to the birthday problem. Starting at $e^{-\frac{k(k-1)}{2m}} \leq 0.5$ i used $ln(x)$ on both sides and multiplied by $-2m$ to get $k(k-1) \geq \ln2*2m$ According to my ...
0
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1answer
54 views

The diophantine equation $a^7+b^7=7^c$

Determine all the triples of positive integers $a,b,c$ such that $a^7+b^7=7^c$.
1
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1answer
32 views

If $a+bi$ is in $E_k$ then $a-bi$ is also in $E_k$?

I'm currently studying the properties of the Motzkin sets $E_k$, $k\in\mathbb{N}\cup\{0\}$ of the ring $\mathbb{Z}[i]$. The definition of $E_k$ is as follows: $E_0=\{0\}$, $E_1=$units of ...
0
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2answers
49 views

Understanding why the public exponent $e$ is chosen the way it is in RSA

I am trying to get a better understanding of RSA. At the moment I am unable to understand the difference between the correctly chosen value of the public exponent $e$ and other possibilities ...
2
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0answers
36 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
votes
2answers
41 views

Solving Linear Congruences With The Euler Totient Functio

I've been asked to calcualte $15^{123456789012345}$ mod $2500$. Now I worked out $\varphi(2500)=1000$ but I can't use Euler's theorem here because $1000$ and $2500$ aren't coprime. Can anyone offer ...
1
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1answer
49 views

Every element of field $F_q$ has $k$th root if and only if $\gcd(q-1,k)=1$

Help me please to prove that: For any $k \in \mathbb{N}$ each element of field $F_q$ is the $k$-th power of some element from that field if and only if $GCD(q-1, k)=1$. My approach Let's look ...
1
vote
1answer
29 views

Solving Linear Congruences With Euler Totient Function

I've been asked to solve the following congruence $x^{1667}\equiv2$ $mod$ $2500$. Am I right in saying there's no solution modulo 2500 to this congruence since even though 1667 is coprime to ...
0
votes
0answers
31 views

Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
3
votes
1answer
49 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
0
votes
0answers
32 views

Number theory and Group theory [on hold]

Can you give me any task which contains Number theory and Group theory?