Some Simple Prime Number Proofs
The number 2 is the only even prime number.
Suppose there is a prime number that is even other than 2. That prime number must not be divisible by 2 or else it would not be prime since 2 is a prime number. However, all even numbers are divisible by 2. Hence, as a result there can not be a prime number that is even other than 2. In other words, 2 is the only even prime number.
The prime numbers 2 and 3 are the only prime numbers that are consecutive.
Suppose that the prime numbers 2 and 3 are not the only prime numbers that are consecutive. Then there exists two prime numbers p1 and p2. Where p2=p1+1 and p1>2. However, since all number are either even or odd and, in particular, following a sequence such as e,o,e,o or o,e,o,e depending on the starting point of the sequence.
Hence, we know that either p1 is odd and p2 is even or p2 is odd and p1 is even. However, if either p1 or p2 is even we know it can not be a prime since it would be divisible by 2, a prime number. Hence, both p1 and p2 can not be prime. Hence at most one of p1 and p2 is prime.
This proves we can not have consecutive primes greater than 2. More clearly 2 and 3 then are the only prime numbers that are consecutive.