Every Prime Number Larger Than 3 Is 1 Away From a Multiple of 6: A Proof
Prime numbers one away from a multiple of 6
It may seem surprising, but every prime number, with the exception of the first two primes, 2 and 3, is within 1 of a multiple of 6. It doesn't matter how big the prime is, it will always be either a multiple of 6 plus 1 or a multiple of 6 minus 1.
For example, let's start with a small prime, 37. 37 = 36 + 1, with 36 being a multiple of 6. We can pick a larger prime such as 839. This equals 840 − 1 where 840 is a multiple of 6 (140 × 6 to be precise). Try to think of some prime numbers yourself; they will always be 1 away from a multiple of 6.
We could keep on checking examples and always getting this result, but as there are an infinite number of primes, doing this won't prove anything. Let's have a look at how we can prove this formally.
For example, let's start with a small prime, 37. 37 = 36 + 1, with 36 being a multiple of 6. We can pick a larger prime such as 839. This equals 840 − 1 where 840 is a multiple of 6 (140 × 6 to be precise). Try to think of some prime numbers yourself; they will always be 1 away from a multiple of 6.
We could keep on checking examples and always getting this result, but as there are an infinite number of primes, doing this won't prove anything. Let's have a look at how we can prove this formally.
What is a prime number?
The first thing we need to do before delving into the proof is to look at what exactly a prime number is.
A prime number is a whole number with exactly two factors: one and itself.
For example, 8 is not a prime number as it has four factors: 1, 2, 4 and 8. Meanwhile, 7 is a prime as its only factors are 1 and 7.
One important thing to take note of from the definition is that 1 is not a prime number as it has only one factor, itself, as opposed to the two needed for a prime.
A prime number is a whole number with exactly two factors: one and itself.
For example, 8 is not a prime number as it has four factors: 1, 2, 4 and 8. Meanwhile, 7 is a prime as its only factors are 1 and 7.
One important thing to take note of from the definition is that 1 is not a prime number as it has only one factor, itself, as opposed to the two needed for a prime.
How to prove our statement
We are now going to look at proving our statement.
To set up our proof, we need to think of all numbers in relation to multiples of 6.
If we take any multiple of 6, we can write it as 6n, where n is a whole number. E.g. 42 = 6 × 7, 126 = 6 × 21 etc.
If we take our 6n and add another 6, we will get the next multiple of 6, which can therefore be written as 6n + 6.
We can see from our definition that 6n and 6n + 6 cannot be prime as they must both be divisible by 6 as well as 1, and 6 is itself divisible by 2 and 3. They must, therefore, have more factors than just 1 and themselves.
Now we have our two consecutive multiples of 6; we can write all of the numbers in between them in terms of 6n. These will be 6n + 1, 6n + 2, 6n + 3, 6n + 4 and 6n + 5, as can be seen in the picture below.
To set up our proof, we need to think of all numbers in relation to multiples of 6.
If we take any multiple of 6, we can write it as 6n, where n is a whole number. E.g. 42 = 6 × 7, 126 = 6 × 21 etc.
If we take our 6n and add another 6, we will get the next multiple of 6, which can therefore be written as 6n + 6.
We can see from our definition that 6n and 6n + 6 cannot be prime as they must both be divisible by 6 as well as 1, and 6 is itself divisible by 2 and 3. They must, therefore, have more factors than just 1 and themselves.
Now we have our two consecutive multiples of 6; we can write all of the numbers in between them in terms of 6n. These will be 6n + 1, 6n + 2, 6n + 3, 6n + 4 and 6n + 5, as can be seen in the picture below.
Numbers written in terms of 6n
With a bit of thought, we can see that all whole numbers are either a multiple of 6 or a multiple of 6 plus 1, 2, 3, 4, or 5. Therefore, every whole number can be written as 6n, 6n +1, 6n + 2, 6n + 3, 6n + 4 or 6n + 5 for some whole number n.
Let's look more closely at these.
We have already seen that 6n itself cannot be prime, but what about the others?
6n + 2 can be factorised to give us 6n + 2 = 2(3n + 1). As this is a whole number (3n + 1) multiplied by 2, then it must have 2 as a factor; hence, it cannot be prime.
In a similar fashion, 6n + 4 = 2(3n + 2), so again, it cannot be a prime number.
The number in the middle of our list is 6n + 3. This equals 3(2n + 1); hence, it has 3 as a factor and also cannot be a prime number.
The two remaining numbers in our list are 6n + 1 and 6n + 5. These do not factorise and, hence, have no obvious factors. In practice, they might have a factor, e.g., 6 × 4 + 1 = 25, which has the factor 5, but we cannot show a definite factor for all n in the way we can for 6n + 2 and the others. This means that 6n + 1 and 6n + 5 are the only numbers on the list that can possibly be prime.
Let's look more closely at these.
We have already seen that 6n itself cannot be prime, but what about the others?
6n + 2 can be factorised to give us 6n + 2 = 2(3n + 1). As this is a whole number (3n + 1) multiplied by 2, then it must have 2 as a factor; hence, it cannot be prime.
In a similar fashion, 6n + 4 = 2(3n + 2), so again, it cannot be a prime number.
The number in the middle of our list is 6n + 3. This equals 3(2n + 1); hence, it has 3 as a factor and also cannot be a prime number.
The two remaining numbers in our list are 6n + 1 and 6n + 5. These do not factorise and, hence, have no obvious factors. In practice, they might have a factor, e.g., 6 × 4 + 1 = 25, which has the factor 5, but we cannot show a definite factor for all n in the way we can for 6n + 2 and the others. This means that 6n + 1 and 6n + 5 are the only numbers on the list that can possibly be prime.
Summary
To summarise:
- We have shown that all numbers can be written as one of 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4 and 6n + 5 for some whole number n.
- 6n, 6n + 2, 6n + 3 and 6n + 4 always have extra factors regardless of the value of n; hence, they can never be prime numbers.
- The only positions that can possibly be prime are 6n + 1 (a multiple of 6, plus 1) and 6n + 5 (the next multiple of 6, minus 1)
- Therefore, all prime numbers are within 1 of a multiple of 6.
Comments
Don't forget to leave your comments below.