1. What are prime numbers?

Each of the natural numbers (positive integers) can be put into one of three directories, according to how many factors /divisors it has.

Given a number N, a divisor is an integer number d such that d divides N evenly with zero remainder. For example, 5 divides 10, or 10 ÷ 2 = 5 with zero remainder. A divisor is also called a factor as we can write 10 = 2 × 5.

Number 1, which is the beginning of numbers, is itself a directory. It has only one divisor, which is 1.

An integer like 5 has only two divisors: one and itself. Any number in this directory is called a *prime number*.

All other numbers must have at least three divisors: one, itself and one proper divisor (which is neither 1 nor itself). Any number in this directory is called a *composite number *.

Examples:

Numbers 2, 5, 7 are prime numbers (also called primes).

Number 4, 6, 9, 12 are composite numbers.

Number 1 is neither prime nor composite numbers.

1a) Please remember that 1 is not a prime!

2. It is helpful to remember a list of smaller primes, like:

2, 3, 5, 7, 11, 13, 17, 19

(in total 8 primes that’s under 20)

We’d like to list some interesting facts and properties about primes, as below:

3. The last digit (units-place) of a prime CANNOT be any digit from 0, 2, 4, 6, 8 (except prime 2 itself); neither can it be digit 5 (except prime 5). Only numbers with last digit 1,3,7,9 are possible candidates of prime numbers 1, 3, 7, 9 (besides 2 and 5, which are exceptions).

4. About 1/4 of the numbers in the first hundred (1-100) are primes.

Among the first hundred,

there are 10 primes that ends with either digit 1 or digit 9;

there are 13 primes that ends either either 3 or 7; they are:

3, 13, 23, 43, 53, 73, 83 (ends in digit “3”)

7, 17, 37, 47, 67, 97 (ends in digit “7”)

And 2 and 5 are primes. To summarize, there are 25 primes in the first hundred.

To look a bit deeper, let’s look at the a few questions as follows.

- Can any composite number be written as the product of primes?
- Are there finitely many prime numbers or infinitely many primes? (“Infinite” means that we cannot count all primes until the end: no matter how many we’ve counted, there are always some more ..)
- Is there a prime-generating math machine? (like a function we input any number, the output is always a prime)

These more amazing stuff with prime numbers! (to be posted)