COMS Tutorial Corner

Prime Numbers – what are they and what’s amazing in them?

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)

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