Do prime number go on forever?

Answer 1

Yes, in the sense that there can be no "highest prime". Here's a simple (though not rigorous) proof.

Suppose there was a highest prime N.

Then N x N' (the next highest prime) x N'' (the next highest prime after that) x ... x 5 x 3 x 2 = some number L. L+1 cannot be divided by N; there would be 1 left over. Similarly, it cannot be divided by any of the other, smaller primes, because there would always be a remainder of 1. L + 1 is therefore a higher prime than N, but we started by assuming there could be a highest prime N. That leads to a contradiction; therefore, there is no highest prime. We don't need to know what L + 1 is exactly, or even what the complete list of primes are, to see that it obviously has to be prime.

(You can try it for small L. For example, 3 x 2 = 6, 6 + 1 = 7, 7 is prime. 5 x 3 x 2 = 30, 30 + 1 = 31, 31 is prime. And so on. Note that this formula doesn't give us every prime; it skips 5, for example, and all the primes between 7 and 31.)