Abstract Nonsense

Chinese Remainder Theorem (Pt. III)

Point of Post: This is a continuation of this post.

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September 6, 2011

The Chinese Remainder Theorem (Pt. II)

Point of Post: This is a continuation of this post.

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September 6, 2011

The Chinese Remainder Theorem (Pt. I)

Point of Post: In this post we discuss the Chinese Remainder Theorem and some of its ramifications, in particular, the multiplicativeness of the totient function.

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September 6, 2011

Some Facts About The Ring of Algebraic Integers

Point of post: In this post we’ll discuss some very basic facts concerning the ring of algebraic integers that shall become useful in other posts.

Motivation

We’ve previously discussed the set of all algebraic numbers at least to the extent to show that the set of all of them is countable. In this post we shall restrict our attention to the algebraic integers, which are basically the result of considering algebraic numbers which are roots of monic polynomials. Our goal in this post is to give some alternate characterization of the algebraic integers in terms of integral matrices, show they are a subring of $\mathbb{R}$, and show that an eigenvalue of a matrix with algebraic integer entries must be itself an algebraic integer.

March 3, 2011

Neat Number Theory/Discrete Math Problem (Pt. III)

Point of post: Since I found this problem enjoyable, for some odd reason, I thought I’d also generalize the methodology in the first proof of this post to the general form and give another alternate proof. These methods have both their pros and cons. The general proof of the problem in my last post has the advantage that it finds a general formula for $a_n$ (see that post for this definition) but is maybe a non-obvious approach and leaves the answer is an ostensibly computationally intractable form. The  proof in this post is more obvious and ends up with a “nicer” answer but leave you only an “estimate” of $a_n$ for any given $n$. Regardless, it is valuable. And, finally for the sake of logical soundness we prove that the form given in the previous proof and the form given in this one are equivalent.

November 6, 2010

Neat Number Theory/Discrete Math Problem (Pt. II)

Point of post: This is a continuation of this post.

Generalization

We now prove a much more generalized theorem. Namely:

Theorem: Let $n_0\in\mathbb{N}$ with prime factorization $p_1^{\alpha_1}\cdots p_{k}^{\alpha_k}$ be fixed and define

$E_n=\left\{m\leqslant n:(m,n_0)=1\right\}$

From this define

$a_n=\left|E_n\right|$

then

$\displaystyle a_n=n-\sum_{j=1}^{k}(-1)^{j+1}\sum_{S\in S_j}\left\lfloor n\left(\prod_{s\in S}p_s\right)^{-1}\right\rfloor$

and thus

$\displaystyle \lim_{n\to\infty}\frac{a_n}{n}=1-\sum_{j=1}^{k}(-1)^{j+1}\sum_{S\in S_j}\left(\prod_{s\in S}p_s\right)^{-1}$

November 4, 2010

Neat Number Theory/Discrete Math Problem

Point of post: In this post I will discuss  the following simple problem:

” Let $E_n=\left\{m\leqslant n:(m,15)=1\right\}$, then define $a_n=\left|E_n\right|$. Compute $\displaystyle \lim_{n\to\infty}\frac{a_n}{n}$

where $(\cdot,\cdot)$ stands for $\text{g.c.d.}$. We will solve this problem in two ways, the first of which is simple but the second of which generalizes to a much, much more general case which we will discuss in the next post.

November 4, 2010