Prime Addition Problem

Since we know that the sum of those prime numbers is a prime number which is greater than 2 (it is an addition of three consecutive prime numbers and 2 is the smallest prime number) and that none of $P_{n}$ , $P_{n+1}$ , $P_{n+2}$ is 2 because the sum of 3 and 5 is even and 2 is even as well which means that $P_{n}+P_{n+1}+P_{n+2}=2k$ (in case of $P_{n}=2$ ) and even numbers greater than 2 are not primes.

The case of $P_{n}=3$ is $3+5+7$ and that is equal to $15$ and $15$ is not a prime number.

The case of $P_{n}=5$ is $5+7+11$ which is equal to $23$ which is a prime!

So $P_{n}=5$ , $P_{n+1}=7$ , $P_{n+2}=11$ and $P_{\alpha}=23$ .

$P_{\alpha}+P_{n}=23+5=28$

Which is the smallest possible solution!