Articles of number theory

Prove that two consecutive square free numbers can have arbitrarily large gaps

I want to the following theorem: The two consecutive square free numbers have arbitrarily large gaps. I tried to prove through definition directly and tried to find a lower bound for the gap of two consecutive square free numbers. However, I failed in this way since I did not figure out it. Does anyone have […]

$\mathbb{Q}_2$ and its quadratic field extensions

How to prove that there are seven non isomorphic quadratic field extensions of $\mathbb{Q}_2$? Approach: I already proved that an unit in $\mathbb{Z}_2$ is congruent with $1,3,5,7 \mod 8$. (maybe that’s useful)

gcd as positive linear combination

Good evening, I have a question concerning the euclidean algorithm. One knows that for $a_1 , \ldots , a_n \in \mathbb{N} $ and $k\in \mathbb{N} $ there exist some $\lambda_i \in \mathbb{Z}$ such that : $$\gcd(a_1, \ldots, a_n) = \frac{1}{k}\sum_{i=1}^n \lambda_i a_i$$ Here is my question: can one find a $m_0 \in \mathbb{N}$ that for […]

If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$?

How to prove that: $\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number For example: $\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$ I know that Knuth showed that: $\gcd(2^{a}-1,2^{b}-1)=2^{\gcd(a,b)}-1$ so: $\gcd(a,b)=1\Rightarrow \gcd(2^{a}-1,2^{b}-1)=1$ but I don’t see whether this fact is useful.

Proofs from the BOOK: Bertrand's postulate Part 3: $\frac{2}{3}n<p \leq n \rightarrow$ no p divides $\binom{2n}{n}$

I have a very hard proof from “Proofs from the BOOK”. It’s the section about Bertrand’s postulate, page 9: I have to show, that for $\frac{2}{3}n<p \leq n$ there is no p which divides $\binom{2n}{n}$. I know $$\binom{2n}{n}=\frac{(2n)!}{n!n!}$$ and from $\frac{2}{3}n<p \leq n$ I follow $3p>2n$. Then $(2n)!$ has only the prime factors $p$ and […]

A power Diophantine equation $(p+q)^n-p^n-q^n+1=m^{p-q}$

Suppose $p$ and $q$ are prime numbers, $n>1$ and $m>1$ are positive integers. Solve the following Diophantine equation:$$(p+q)^n-p^n-q^n+1=m^{p-q}$$I made this problem and I was trying to find $p$ and $q$ first. Looking mod $p$, $q$ and $p+q$ gives$$m^{p-q} \equiv 1 \pmod {pq(p+q)}$$Does this provide useful information about $p$ and $q$?

Convergence of Riemann Zeta Function for Normed Arguments

My question is concerned with the convergence of the sum $$\displaystyle \sum_{\substack{\ \ m \in \mathbb{Z}^d} \\ {\ \ \ m \neq 0}} \frac{1}{\|m\|_{d}^s},$$ where $d \in \mathbb{N}$, and $\|m\|_d := \sqrt{|m_1|^2 + … + |m_d|^2}$ denotes the usual Euclidean norm on $\mathbb{R}^d$. Is it true that this sum converges whenever $\mathrm{Re}(s) > d$? Obviously, […]

The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If $\frak{P}$ is a non-zero prime (maximal) ideal of $B$ is it true that as $\sigma$ runs through $G$ the […]

Identity, Bernoulli number

I am trying to prove the following for a very, very long time: $$ 2k=\sum_{j=1}^k \binom{2k+1}{2j}2^{2j}B_{2j} $$ Here, $B_{2j}$ are Bernoulli numbers. I would be extremely happy if somebody could help me with this!

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} – y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out a proof and this turns out to be true from $n=1$ to $n=20$. After that I didn’t try, but I would like […]