The Joys of Mathematics

One of the great joys in mathematics is the ability to use quite simple mathematics to prove some very deep truths. One of my favourite examples of this involves the prime numbers. As most people know, a prime number is one which can only be divided by itself and 1. The first few examples are 2, 3, 5, 7, 11, … As you can see, prime numbers are initially very common, but as numbers get larger, they get increasingly scarce.

4 of the first 10 integers are prime, an average of 40%

25 of the first 100 integers are prime, an average of 25%

168 of the first 1000 integers are prime, an average of 16.8%

78,498 0f the first 1,000,000 integers are prime, an average of 7.89%

50,847,534 of the first 1,000,000,000 integers are prime, an average of 5.08%

So, a natural question is, “do the primes go on for ever?” or is there a finite number of primes.

While studying GCSE mathematics, we come across the fact that any number can be represented as a product of prime numbers. So, for example,

35 = 5 X 7,    60 = 2² X 3 X 5,   525 = 3 X 5² X 7

Now, consider the following calculations

If we multiply together the first 2 primes and add 1, 2 X 3 + 1 = 7. The result, 7, cannot be divided by either 2 or 3.In fact 7 is a new prime.

If we multiply together the first 3 primes and add 1, 2 X 3 X 5 + 1 = 31. The result, 31, cannot be divided by any of 2, 3 or 5. In fact 31 is a new prime.

If we multiply together the first 4 primes and add 1, 2 X 3 X 5 X 7 + 1 = 211. The result cannot be divided by 2, 3, 5 or 7 and is a new prime.

If we multiply together the first 5 primes and add 1, 2 X 3 X 5 X 7 X 11 + 1 = 2311. The result cannot be divided by 2, 3, 5, 7 or 11 and is a new prime.

If we multiply together the first 6 primes and add 1, 2 X 3 X 5 X 7 X 11 X 13 + 1 = 30031. Again this result cannot possibly be divided by 2, 3, 5, 7, 11 or 13. On this occasion 30031 is not a new prime, however 30031 = 59 X 509 which are both new primes.

So now to our proof that there are an infinite number of primes.

STEP 1 – Assume that there are a finite number of primes p₁, p₂, p₃, …. p_n

STEP 2 – Construct the number  N = p₁ X p₂ X p₃ X …. X p_n + 1

As above, N cannot be divided by any of the primes p₁, p₂, p₃, ….. or p_n

So N is either a new prime, or is the product of new primes

This contradicts our statement In STEP 1

So the statement in STEP 1 cannot be true.

Hence, the number of primes is infinite.