“Is there a prime number next?” – this is the question that many mathematicians are asking themselves. Some people believe that every number, with the exception of 1 and 2 are prime numbers. This may be true to some extent, but it isn’t always easy to tell which numbers are even or odd in order to determine whether they are prime or not. In this post we will discuss what factors make a number “prime,” as well as how you can figure out if it is prime by using simple mathematics!
What are prime numbers?
Prime numbers are any integers that can be divided evenly by only one and itself. For example, the first nine natural numbers (the counting numbers) are divisible by both themselves and one:
So in this case we have a total of eleven even/odd number pairs! What would happen if you added another single integer to the list? Let’s find out…
This time there is twelve odd/even number pairs which means all but two of them are prime. With every additional integer on our list it becomes more difficult to determine whether or not they are prime because with each new member comes an increased chance for these quantities to “pair up” as being either even or odd. You may ask yourself, “what are the odds that a number will be prime?” or if there’s even a next one.
The chances of any integer being prime are about 50%. This is because when you start with two numbers and keep randomly adding integers to them until only one remains, each time it happens half of the time!
So what does this all mean for us? For starters, we can rest easy knowing that every odd number from three onwards is definitely going to be a prime. Second… well I’ll leave that up to your imagination ;).
Happy Maths Hunting!
We’ve found out about how many pairs were left as an even/odd pair after starting off with eleven integers:
11 – 11 = 0 which means that there are no pairs left.
We’ve found out about how many pairs were left as an even/odd pair after starting off with twenty-one integers:
21 – 21 = 0 which means that there are no pairs left.
The chances of any integer being prime are 50% because when you start with two numbers and keep randomly adding integers to them until only one remains, it happens half of the time! For example if we started with a number like thirteen then every other odd number up to seventeen would be a prime (i.e., twenty-one is not). If we started with five or twelve starting off with thirty three integers: 33 – 33 = 0 which means that there are no pairs left. We’ve found four sets of “even” numbers that are not prime.
If this pattern continues then for all ƒ(x) ≥ 34 must exist an even x such that ƒ(x) + ƒ(x) = 0.
This may seem obvious if we’ve already seen that every odd number greater than or equal to 17 is prime, but what about the integers less than seventeen? The only way for this pattern to be true (that all even numbers are not prime and every other set of two consecutive integers leads to a pair of primes up until there are no more pairs left), would mean that any integer can be divided by two with an answer greater than or equal to one. That means that it’s possible for our first number in the sequence, which was thirteen, could also have a divisor between eleven and thirteen! It seems implausible at first glance because you need to go to the square root of both eleven and thirteen are less than one, which would make them composite numbers. But this is when we can start using modular arithmetic for those smaller numbers!
The pattern still holds true with no exceptions up until there are only two consecutive integers left: 17 and 19. Now what? Well if you’ve made it this far in math class then you might already know that all prime numbers “greater” than seventeen have a remainder of zero after being divided by two. The last step in finding out whether or not every integer greater than or equal to seventeen is even and an odd number leads to a pair of primes is checking the answer for 17+19 = 36 as our final test case (17*19).
So are all odd numbers prime? It turns out that the answer is “no, some are”. To prove this theorem it looks like we need to compare any number greater than or equal to seventeen with its reciprocal. If they are both even and an integer then these two will be a composite number! And if either one of them have a remainder after dividing by two, then they’re not prime. So now you know what’s going on in math class today :)
The Pattern Still Holds True With No Exceptions Up Until There Are Only Two Consecutive Integers Left: 17 & 19
But When We Can Start Using Modular Arithmetic For Those Smaller Numbers!