The Infinite Prime Number Proof

The Infinite Prime Number Proof

         As the title states, this blog will be primarily concerned with Euclid's 2500 year old proof which we began in class. Proving such a statement required us to prove the contradiction false which we began by assuming n to be an element of the natural numbers. So essentially, the set of prime numbers must be less than or equal to n. We then set m equal to the product of all the elements of P (the set of prime numbers) and deduced that m + 1 could then be a prime number. Now we have to take into account that m + 1 may or may not be a new prime number. So from here my thought process was that if it is a new prime number, this side of the proof is resolved. On the other hand if its not a new prime number then we have to somehow prove that a new prime number somehow arises from this. Now if we factor any integer we can rewrite that integer as the product of prime numbers. So if we take the number m + 1, it cannot be divided by any element of P otherwise there would be a remainder of 1. Therefore in order to write m + 1 as a product of some prime number times some other number, that prime number must be a new prime number not listed in P. To conclude, in both cases we result in new prime numbers outside of the set P and so there must be an infinite amount of prime numbers.
         Before I end the blog I'd like to state a quick thought on our recent midterm. I thought it was very straight forward and was pleasantly surprised at its difficulty. For a 60 minute test I believe it was very fair. To conclude, I'm really liking this course and aim to take more courses of the like in the future.