1. We first prove there are infinitely many primes.
2. Assume the number of primes is finite.
3. Thus, there exists a number that is the product of all primes.
4. Thus, there exists a number that is one more than the product of all primes, let us call this number A.
5. A is larger than any prime, thus different from any prime, thus not a prime.
6. Thus, there is at least one prime that A can be divided by.
7. But by it's very definition as the product of all primes + 1, the remainder of dividing this number by any prime is 1.
8. Thus, A is divisible by no prime, and prime.
9. Contradiction follows from assumption 2, thus assumption is false.
10. There are thus infinitely many primes.
11. There are finitely many number below 10^10^10^10^10
12. Thus, there must at exist infinite primes above 10^10^10^10^10
I don't know any of these numbers, but showing that a contradiction can be derived from assuming that such a number not exist is the common way to do it.
1. We first prove there are infinitely many primes.
2. Assume the number of primes is finite.
3. Thus, there exists a number that is the product of all primes.
4. Thus, there exists a number that is one more than the product of all primes, let us call this number A.
5. A is larger than any prime, thus different from any prime, thus not a prime.
6. Thus, there is at least one prime that A can be divided by.
7. But by it's very definition as the product of all primes + 1, the remainder of dividing this number by any prime is 1.
8. Thus, A is divisible by no prime, and prime.
9. Contradiction follows from assumption 2, thus assumption is false.
10. There are thus infinitely many primes.
11. There are finitely many number below 10^10^10^10^10
12. Thus, there must at exist infinite primes above 10^10^10^10^10
I don't know any of these numbers, but showing that a contradiction can be derived from assuming that such a number not exist is the common way to do it.