Prime Squares: $(a,b,c,d)$ such that $a,b,c,d,a+b+c+d$ are each between twin primes

Question

[Edited since answer] I have an updated and more advanced version of the prime square in this post: Prime Square: Updated Concept

Hello everyone! I would like some feedback on a new idea of mine.

It's called a Prime Square.

This is what it looks like:

• Definition: Edit since answer A Prime Square is composed of four numbers, each between a pair of Twin Primes (18, 18, 18, 18) AND the sum of all four numbers is also between a pair of Twin Primes(72).

The Prime Square has been found! 18+18+18+18=72

Answer 1

If you let numbers repeat, $618+2238+2238+2238=7732$ works.

If not, $6+2238+2688+3360=8292$.

If you want all numbers the same, $18+18+18+18=72$ is the smallest ($60$, $108$, and $150$ are the others that I found, up to 250).

Computed in Magma:

N := {i : i in [1..1000] | IsPrime(i-1) and IsPrime(i+1)};
{{a,b,c,d} : a,b,c,d in Numbers | a+b+c+d in Numbers};`