Articles of number theory

How to prove that the partial Euler product of primes less than or equal x is bounded from below by log(x)?

How does one prove $\prod_{p \leq x}(1 – \frac{1}{p})^{-1} \geq \log(x)$?

Every quotient of a number ring is finite

Let $K$ be a number field, i.e. a subfield of $\mathbb{C}$ of finite degree over $\mathbb{Q}$. Let $\mathscr{O}_K$ be the ring of integers of $K$, i.e. algebraic integers which are in $K$. Let $I$ be an ideal of $\mathscr{O}_K$. I read many times that the quotient $\mathscr{O}_K/I$ is obviously/clearly a finite ring, but i’ve never […]

Is this it a theorem that numbers congruent modulo two relatively prime $m,n$ are congruent modulo $mn$?

“If $a \equiv b \pmod m$ and $a \equiv b \pmod n$ and $\gcd(m,n)=1$, then $a \equiv b \pmod {mn}$ “ Is that a true theorem? I can’t find it in my textbook!

The number of partitions of $n$ and the $n$th Fibonacci number.

I’m very sorry if this is a duplicate in any way. There’s a lot of material out there on connections between these sequences so it’s a possibility . . . Let $P_n$ be the number of partitions of $n$ and $F_n$ be the $n$th Fibonacci number. I’ve been learning how to use GAP. Naturally, the […]

How do I find(isolate) the n-th prime number?

So I wanted to solve this SPOJ problem and I did some research about finding the n-th prime number. This formula came across and it stated that the n-th prime must be in this range: $n \ln n + n(\ln\ln n – 1) < p_n < n \ln n + n \ln \ln n$ for […]

In a $p$-adic vector space, closest point on (and distance from) a plane to a given point?

Let $\| x \| =\sqrt{x^T x}$ be the Euclidean norm on $\mathbb{R}^n$. Consider the point $z \in \mathbb{R}^n$, and the plane $P = \{x \in \mathbb{R}^n : a^T x = b\}$ where $0 \neq a \in \mathbb{R}^n$, $b \in \mathbb{R}$. Orthogonal projection gives the point \begin{align}\label{1}\tag{1} y = z – \frac{(a^T z – b)}{a^T a} […]

How to check if any subset of a given set of numbers can sum up to a given number

Given a number, say $x$, and a set of numbers made up of only $k$ different numbers, where each of the $k$ numbers is repeated $n_1,n_2,\dots n_k$ times. How do I tell if it is possible to find a subset such that it sums to $x$. E.g.: $$x=6 , k=3$$ $$S=\{1,2,3,3,2,1\}$$ $$n_1=2, n_2=2, n_3=2$$ $$1+2+3=6$$ […]

Prime divisibility

I have the following assertion in my notes from last year that I’m trying hard to digest, but I think it isn’t true: If $p$ is prime $\Leftrightarrow$ if $p | ab$ then either $p | a$ or $p | b$ or both. A valid proof must prove both directions, so: if $p$ is prime, […]

Generating Coprime Integers

Is there a formula for generating a set of Coprime integers that every element of this set is coprime to the other elements in this set? I want to create a collection of this formulas!

Are there groups with conjugacy classes as large as the divisors of perfect numbers?

Are there groups, where the conjugacy classes have elements according to the divisors of perfect numbers larger than $6$? I already got the first example, so therefore $6$ is excluded: $S_3$. The corresponding GroupProps page on groups of order $28$ is still empty, but OEIS/A000001 says that there are $4$ groups having $28$ elements. Any […]