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

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 […]

“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!

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 […]

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 […]

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} […]

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.: $$x=6 , k=3$$ $$S=\{1,2,3,3,2,1\}$$ $$n_1=2, n_2=2, n_3=2$$ $$1+2+3=6$$ […]

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, […]

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, 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 […]

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