Although 1 is only divisible by itself and "1", seemingly meeting the definition of a prime, it is by convention excluded from the set of primes.

I suppose this is because of all its additional special properties. If you were to count 1 as a prime factor in a greater number, how many occurrences of it would you give this factor? An infinite number would be possible.

3 is the second prime, not the third.

Although 1 is only divisible by itself and "1", seemingly meeting the definition of a prime, it is by convention excluded from the set of primes.

I suppose this is because of all its additional special properties. If you were to count 1 as a prime factor in a greater number, how many occurrences of it would you give this factor? An infinite number would be possible.