Is There a Biggest Prime Number or Do They Continue Infinitely?
What Are Prime Numbers?
Prime numbers are positive whole numbers that have exactly two factors (numbers that divide into them perfectly with no remainder).
For example, 2 is a prime number, as the only numbers which will divide it perfectly are 1 and 2. A larger example of a prime number is 29. With a bit of checking, we can see that the only factors are 1 and 29.
On the other hand, 8 is not a prime number as it is divisible by 1, 2, 4 and 8, hence has four factors. 1 is also not a prime as it has exactly one factor: itself.
We can also see from our examples that the two factors of a prime number will always be 1 and itself.
One question that often comes to mind when people first come across the concept of prime numbers is whether or not the largest prime exists. It can certainly feel like if we take a number large enough, there must be something that divides into it. Let's see if we can prove this one way or the other.
For example, 2 is a prime number, as the only numbers which will divide it perfectly are 1 and 2. A larger example of a prime number is 29. With a bit of checking, we can see that the only factors are 1 and 29.
On the other hand, 8 is not a prime number as it is divisible by 1, 2, 4 and 8, hence has four factors. 1 is also not a prime as it has exactly one factor: itself.
We can also see from our examples that the two factors of a prime number will always be 1 and itself.
One question that often comes to mind when people first come across the concept of prime numbers is whether or not the largest prime exists. It can certainly feel like if we take a number large enough, there must be something that divides into it. Let's see if we can prove this one way or the other.
The Fundamental Theorem of Arithmetic
One thing we will need for our proof is the Fundamental Theorem of Arithmetic. This states that any positive whole number can be written as a unique product of prime factors. For example 12 = 2 × 2 × 3 and 21 = 3 × 7. The formal proof of this is beyond the scope of this article, but it doesn't take too much investigation to be comfortable that this theorem holds. For more information on prime factors read my article on writing a number as a product of prime factors.
The important thing to take from this for us, is that any non-prime number must have prime factors.
The important thing to take from this for us, is that any non-prime number must have prime factors.
Euclid and His Work on Prime Numbers
In this article, the maths we use is based on a similar proof used by the ancient Greek mathematician Euclid in his ground-breaking work, Elements, from circa 300 BC.
Euclid is considered the 'father of geometry,' and Elements laid the groundwork for a lot of modern mathematics. It also contained a significant amount of rigorous work on number theory, including his work on prime numbers.
Euclid is considered the 'father of geometry,' and Elements laid the groundwork for a lot of modern mathematics. It also contained a significant amount of rigorous work on number theory, including his work on prime numbers.
Is There a Largest Prime?
Let's assume that there is a finite list of prime numbers, which we shall call p1, p2, p3, …, pn where pn is the largest prime.
We will now define P as the product of all of these prime numbers so:
P = p1p2p3...pn
We will now create a new number q, which is one more than P:
q = P + 1
We now face a dilemma.
If q is itself prime, then being one more than the product of all of the prime numbers in our list, it must be larger than any of them and is the new largest prime number. This contradicts pn being the largest prime.
If q is not a prime number, then it must have prime factors, as mentioned earlier. However, as q is one more than the product of all of the primes in our list, then it cannot be a multiple of any of them. (We can see this because if we divide q by any of p1 to pn, we will be left with the 1 as a remainder). Therefore, prime factors outside of our list must exist. But we defined our list as containing all of the prime numbers. Again, we have a contradiction to our original definition.
This is called proof by contradiction. By showing that our original assumption will always lead to a contradiction, we have shown that the original assumption must be false. In this case, we have shown that the list of prime numbers is infinite and hence no largest prime number exists.
We will now define P as the product of all of these prime numbers so:
P = p1p2p3...pn
We will now create a new number q, which is one more than P:
q = P + 1
We now face a dilemma.
If q is itself prime, then being one more than the product of all of the prime numbers in our list, it must be larger than any of them and is the new largest prime number. This contradicts pn being the largest prime.
If q is not a prime number, then it must have prime factors, as mentioned earlier. However, as q is one more than the product of all of the primes in our list, then it cannot be a multiple of any of them. (We can see this because if we divide q by any of p1 to pn, we will be left with the 1 as a remainder). Therefore, prime factors outside of our list must exist. But we defined our list as containing all of the prime numbers. Again, we have a contradiction to our original definition.
This is called proof by contradiction. By showing that our original assumption will always lead to a contradiction, we have shown that the original assumption must be false. In this case, we have shown that the list of prime numbers is infinite and hence no largest prime number exists.
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