More prime numbers!!

Haven't posted for a while (not that anybody reads this...). Gonna try to make this one more formal looking, too.

Find the least positive integer n such that for each prime p, p^2 + n is not a prime number.

All primes, except for 2, are odd. If they were even, then they would be divisible by 2, which means they are not prime.

Any number which is even, except for 2, is not prime. This is implied from the previous statement.

An odd number squared will be an odd number, or, an odd number times an odd number is an odd number. This is common knowledge.

An odd number plus an odd number will be an even number. This is common knowledge.

Any prime number, except for two, when squared and added to an odd number, will not be prime. This can be inferred from the previous statements. Any prime number, except for two, when squared is odd, and any odd plus another odd is even, and any even number except for two is not prime.

Casework, working upwards from 1, to find the smallest possible value of n:
N is 1:
2^2+1=5, a prime number.

N is 2:
3^2+2=11, a prime number

3:
2^2+3=7, a prime number

4:
3^2+4=13, a prime number

5:
2^2+5=9, NOT prime number

Also, since 5 is odd, then any other prime number, besides 2, squared plus 5, (p^2 + 5), will also not be prime. Therefore, n = 5.