Prime numbers

Eucild was the first to prove that prime number is infinity because it goes on forever and ever. He used the well known prime number to add it to every product. For example, 2*3+1=7. Then he continue by adding 5 to the same problem, 2*3+5+1=31 and it came out to be a prime number too. Unfortunally some of the problem didn't come out to be a prime number but there are still more prime when he keep doing the process. This is his proof:P1,P2,P3,..Pn representing the prime and the numbers. Multiply them and then add by 1 calling this a new interger q. When hte multipy and add to equal to q it is a prime number. When the answer is not q then it have to be divided by and it's called r. Dividing q with the prime number will come to a remainder 1 so it can not be a prime number.