Articles of number theory

Why the zeta function?

Why is the zeta function, $\zeta(s)$ used to obtain information about the primes, namely giving explict formula for different prime counting functions, when there are many other functions that encode information about primes? For example, the relation: $$\frac{-\zeta'(s)}{\zeta(s)}=\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$$ Could be seen as a special case of a polylogarithm identity: $$\frac{d}{ds}\text{Li}_s(x)=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s}\text{Li}_s(x^n)$$ Where $x=1$.

A fast factorization method for Mersenne numbers

Given a prime number $p$ and a Mersenne number $M=2^p-1$: Is it true for every prime factor $q$ of $M$ that $q\equiv1\pmod{p}$? For example, $p=29$ and $M=536870911=233\cdot1103\cdot2089$: $ 233=29\cdot 8+1$ $1103=29\cdot38+1$ $2089=29\cdot72+1$ If yes, then is there a method which exploits this fact in order to factorize $M$ (or decide it is prime), which is […]

Question related to pseudoprimes and Carmichael numbers

We know by Fermat’s little theorem that if $m$ is a prime then for any $n \in \mathbb{Z}$, then $$ n^m \equiv n (mod \ m). $$ I was wondering if there was a composite $m$ that satisfies this condition? (Maybe this condition is equivalent to being a Carmichael number, but I wasn’t sure…) Thank […]

Finding all positive integers which satisfy $x^2-10y^2=1$

I’m interested in finding positive integers which satisfy an equation. I’ve been thinking about the following equation: $$x^2-10y^2=1\ \ \ \ \ \ \cdots(\star).$$ Then, I’ve just got the following (let’s call this theorem): Theorem: If $(x,y)$ satisfies $(\star)$, then $(20y^2+1,2xy)$ also satisfies $(\star)$. Proof: Letting $x=10n+1$, we get $2n(5n+1)=y^2$. Hence, let’s consider the case […]

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any positive integer n, if n is even, divide it by 2; otherwise multiply it by 3 and add 1. If […]

Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) lower and upper bounds to the following arithmetic / number-theoretic expression: $$\frac{I(x^2)}{I(x)} = \frac{\frac{\sigma_1(x^2)}{x^2}}{\frac{\sigma_1(x)}{x}}$$ where $x \in \mathbb{N}$, $\sigma_1(x)$ is […]

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn’t include the full proofs. That’s fine for the question as it stands, but I find the subject interesting enough to wonder about the details. So here I’m asking. Let’s start by defining […]

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, any results on reverse direction ? If we assume Goldbach Conjecture holds true, can we get any results about GRH ?

Show $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$

Let $\zeta_{p^n}$ be the primitive $p^n$-th root of unity where $p$ is a prime and $K_n=\mathbb Q(\zeta_{p^n})$ the $p^n$-th cyclotomic field. Let $K_\infty=\bigcup K_n$. Could someone give a proof of the isomorphism $\text{Gal}(K_\infty/\mathbb Q)\cong \mathbb Z_p^{\times}$? Many thanks in advance.

Some co-finite subsets of rational numbers

Is there any $A\subseteq \mathbb{Q}$ such that $A-A$ is a non-trivial co-finite subset? Note that $A-A=\{a_1-a_2: a_1,a_2\in A\}$, also see A type of integer numbers set.