Commentary on Proposition IX.20

Back to Proposition IX.20.

We have here the important proposition that the number of prime numbers is infinite.

The proof will be seen to be the same as that given in our algebraical text-books. Let a, b, c, … k be any prime numbers. Take the product abc … k and add unity. Then (abc … k + 1) is either a prime number or not a prime number.

  1. If it is, we have added another prime number to those given.

  2. If it is not, it must be measured by some prime number [VII. 31], say p. Now p cannot be identical with any of the prime numbers a, b, c, … k. For, if it is, it will divide abc … k.

Therefore, since it divides (abc … k + 1) also, it will measure the difference, or unity : which is impossible.

Therefore in any case we have obtained one fresh prime number. And the process can be carried on to any extent.