Atomic Numbers

Prime numbers are arguably the superstars of number theory. They are the building blocks from which we get every other number, as we will see. In this post, we will define prime numbers, prove several known results, and field many, many open problems related to them.

Definition 1. Let $n$ be any integer. We say that $d$ divides $n$ , denoted $d \mid n$ , if there exists an integer $k$ such that $n = kd$ . For other nomenclature, we say that $d$ is a divisor or a factor of $n$ .

It is obvious that $1$ is a factor of any integer $n$ . Furthermore, any number $d$ is a factor of $0$ . It is also obvious that $d \mid n$ if and only if $d \mid -n$ . Henceforth, unless it is useful to consider negatives, we will assume $n, d > 0$ .

We will need several divisibility results in the course of our study of number theory. This is by no means exhaustive, but is certainly useful for our discussions.

Theorem 1. We have the following divisibility properties.

For integers $a, b$ , $d \mid a$ and $d \mid b$ implies $d \mid (a + b)$ .
For integers $a, c, d$ , $d \mid a$ implies $d \mid (ca)$ .
For integers $a, b, c_1, c_2, d$ , $d \mid a$ and $d \mid b$ implies $d \mid (c_1 a + c_2 b)$ .
For positive integers $a , b$ , $a \mid b$ implies $a \leq b$ .
For positive integers $a , b$ , $a \mid b$ and $b \mid a$ implies $a = b$ .

Proof. For the first result, find integers $k_1,k_2$ such that $a = k_1 d$ and $b = k_2 d$ . Then

$a + b = k_1 d + k_2 d = ( k_1 + k_2)d.$

Therefore, $d \mid ( a + b)$ . The second result follows similarly, and the third result arises from repeated applications of the first two results. For the fourth result, find an integer $k \geq 1$ , such that

$b = ka \geq 1 \cdot a = a.$

The final result follows from $a \leq b$ and $b \leq a$ .

Prime numbers are numbers that can only be divided by either $1$ or itself. We formalise their definition as follows.

Definition 2. Let $n \geq 2$ be a positive integer. We say that $n$ is composite if it has at least one divisor $1 < d < n$ . We say that $n$ is prime if it is not composite.

Lemma 1 describes an equivalent formulation of a prime number.

Lemma 1. A positive integer $p \geq 2$ is prime if and only if for any positive integer $d$ , if $d \mid p$ , then $d = 1$ or $d = p$ .

Can all numbers be written as a product of primes? This is the famed fundamental theorem of arithmetic, which we will first state, but only prove later on.

Theorem 2. For any number $n \in \mathbb N$ such that $n \geq 2$ , there exist primes $p_1,\dots,p_N$ such that

$\displaystyle n = p_1 \cdots p_N = \prod_{k=1}^N p_k.$

Proof. We prove this result independently here.

The full theorem asserts a stronger result, that the decomposition into primes is unique. Immediately, we can state a rather surprising result.

Theorem 3. There are infinitely many primes.

Proof. Suppose, for a contradiction, that there are only finitely many primes $p_1,\dots,p_n$ . Consider the number $p := p_1 \cdots p_n + 1$ . Since $p \neq p_i$ for any $i = 1,\dots,n$ , $p$ is not prime.

Therefore, $p$ is composite. By the fundamental theorem of arithmetic, there exists some $p_i$ such that $p_i \mid p$ . Since it is obvious that $p_i \mid (p - 1)$ , we have $p_i \mid (p - (p-1))$ so that $p_i \mid 1$ . Hence, $1 < p_i \leq 1$ , a contradiction.

Thus, we contradicted the claim that there are only finitely many primes. Therefore, there are infinitely many primes.

The first few primes are $2, 3, 5, 7, 11, \dots$ , so on and so forth. The record for the largest known prime keeps getting larger, and has at least $41\, 024\, 320$ digits. A seemingly related question pertains twin primes. A pair $(p_i,p_{i+1})$ of consecutive primes are twin if $p_{i + 1} - p_i = 2$ . For instance, $(3,5)$ , $(5, 7)$ , and $(11, 13)$ are twin primes. The question is simple.

Conjecture 1. There are infinitely many pairs of twin primes.

This is a conjecture, known uncreatively as the twin prime conjecture, since there is no known proof of the result yet. So far, teams of mathematicians have found infinitely many pairs of primes with $p_{i +1} - p_i = k$ , where $k \leq 70\, 000\, 000$ . As of writing this, the current smallest unconditional record for $k$ is $k \leq 246$ . If further mathematical results are proven, the number shrinks further to $k \leq 12$ and even $k \leq 6$ .

Another conjecture seems straightforward to prove, yet due to our collective skill issue, has not. This conjecture is straightforward: observe that $4 = 2 + 2$ , $6 = 3 + 3$ , $8 = 3 + 5$ , so on and so forth. Notice that each even number listed is the sum of two primes. That is the conjecture, known as Goldbach’s conjecture.

Conjecture 2. For any positive even integer $n$ , there exist primes $p_1,p_2$ such that $n = p_1 + p_2$ .

It seems loomingly stunning that as straightforward as our definition of a prime number is, we really don’t know a lot about them. There’s even a one-million-dollar-problem on the table that is intimately connected with locating these primes.

Intuitively, the reason for this difficulty is that a prime number is defined in opposition to a composite number:

A number is composite if it has smaller parts; a number is prime if it does not.
A number is rational if roughly speaking it can be written as a fraction of integers; a number is irrational if it is not rational.
In the more advanced topic of topology, a topological space is disconnected if it can be separated but connected if it is not disconnected.

In general, it is easier to work with mathematical ideas that are explicitly defined, rather than ideas that are implicitly defined (i.e. it is the opposition of something else), and prime numbers are no exception to that general rule.

Welcome to number theory—easy-to-state problems with not-easy-at-all solutions, if any. To discusss discrete mathematics, we will keep the problems relatively solvable.

—Joel Kindiak, 13 Nov 24, 1722H

KindiakMath

Atomic Numbers

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