Definition of divides

As a warning, this whole thing may prove terribly useless. It will especially seem unnecessary for now, but I am hoping it helps with a later problem. Also, we’re just sort of testing out the MathJAX.

Lately, I have been looking into cases for $a,b,n\in\mathbb{N}$ where $$a\mid b^n \implies a\mid b.\tag{1}$$ While I have grown up on the definition of divide being $$a\mid b \iff \exists k\in\mathbb{Z}\, \text{ s.t. }\, a\cdot k = b,$$ this latest quest has led me to seek a new definition. So, I will first try to look at this definition, and then maybe we can look into some of the cases where $(1)$ is true.

First, let’s throw out when $a=1$. Since $1$ divides everything already, we will count this as trivial. We will write $a$ in terms of its prime factorization: $$a=\prod_{i=1}^{\infty}p_i^{\alpha_i}$$
Where each $p_i$ be the $i$th prime number, i.e. $p_3=5.$ To see an example, we are writing the number $12$ as $12=2^2\cdot 3^1\cdot 5^0\cdot 7^0\cdots$ where the multiplication is infinite, but at some point the $\alpha_i$ have become $0$.

Let the factorization of $b$ be written as $$b=\prod_{i=1}^{\infty}p_i^{\beta_i}.$$

Then, we have this different definition of divide: $$a\mid b \iff \alpha_i\leq \beta_i, \forall i$$

Now, looking back at $(1)$, let’s assume $a$ is prime. Then for some $i=k$ $\alpha_k = 1.$ Then notice that $a\mid b^n \implies 1 \leq n\beta_k.$ As we are only considering non-negative, this means that each of $n$ and $\beta_k$ must be $1$ or greater, which tells us that $\alpha_k\leq\beta_k.$ This is the definition of divides, and so $a\mid b,$ giving $(1).$ In the case of $a$ prime, this implication is already well-known. We simply used a different approach.

Indeed, even for composite $a$, if each $\alpha_i\leq 1$ we will find the same result. This gives us the “strongest” case so far, which is that $(1)$ holds for any squarefree $a.$

Summer Time

Oh heck yeah. It’s summer time. I’m tutoring over the summer, so getting to tell everyone about algebra and statistics. I might even get to do some astronomy. Speaking of astronomy, Saturn looked nice and bright last night.

Anyway, I met up with a friend to look over some differential equations. It would be good to stay nice and fresh on that, so we started at the beginning of our textbook again and looked over critical points and direction fields. In the future, I’ll probably type up more about what I learn or show some better illustrations. I’m just trying to get this ball rolling.