The New Oxford Dictionary Of English is wrong! If you read their definition (see above) and treat that definition as correct you are effectively saying that 1 is a prime number, after all 1 is only divisible by itself (1) and unity (1). But one is not a prime number.

So what is the correct definition?

A prime number is a positive integer which has exactly 2 different positive integers that divide it evenly.

Why is it important that 1 is not a prime number?

If you include 1 as a prime number then you run into some problems. For instance, every positive integer that is not prime can be made up my multiplying at least 2 prime numbers together and there's only one way to do it for each number. For instance, 280 = 2 x 2 x 2 x 5 x 7 and 282= 2 x 3 x 47. If you let 1 be a prime there is more than one way to make up a number. If we take 280 as our example we could make it up in the following ways:

I could go on forever but you see the idea, the factorization is no longer unique and this idea is quite important to mathematicians

Furthermore, there are a whole bunch of theorems in Number Theory that tell you something about prime numbers. But most of these theorems just aren't true for the number 1

So in light of these facts, we just declare the number 1 to not be a prime. So that's why we don't WANT 1 to be a prime

What are the other prime numbers and how do I calculate them easily?

There is no known pattern for prime numbers each one has to be checked by counting it's divisors. If there are exactly 2 divisors  it is a prime number.

If you want to find all the prime numbers under a certain number we can use The Sieve of Eratosthenes. I'll demonstrate how it works

Say we want to find all the prime numbers under 100, first we write out the numbers in a grid and cross out 1 as it is not a prime number

now look for the lowest number that is not crossed out, this is a prime number. From the grid we can see that it is 2. Circle 2 (I have coloured it yellow), then cross out every multiple of 2 (e.g. 2,4,6,8,10).

Next repeat the process, the lowest number not crossed out is 3, so mark 3 as prime and cross out 6,9,12,15 etc. (some, like 6 and 12 will have already been crossed out)

the next prime is 5 as it is the lowest number not crossed out so cross out all the multiples of 5

if you repeat this process until all the numbers are either marked as prime or crossed out you now have a list of all prime numbers below 100

We can extend this grid up to 1,000 or 1,000,000 or whatever number we chose, but it can get quite time consuming?

Is there a way to decide if a number is prime or not?

This is called Primality testing.

While simple to describe and understand, the properties of prime numbers are unexpectedly elusive, despite that fact that they have been the subject of intense mathematical inquiry since the days of Euclid and Eratosthenes. Determining whether a given number is prime or composite has proved a difficult problem. This problem has received immense attention in recent years due to the widespread use of prime numbers in encryption algorithms such as the RSA algorithm. For example, if there were efficient algorithms for factoring large numbers, the vast majority of encrypted electronic communications worldwide would be easily crackable.

There are some tests that you can do to prove that a number is not prime but these don't prove that a number is prime.

Take these numbers:

We can confidently say that all the even numbers are not prime as the only even prime is 2 also we can say that any number ending in 5 or 0 is not prime as it will be divisible by 5