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

So … none of the numbers is actually prime, and for large numbers it starts to look distinctly rectangular. (-: — JdeBP, Nov 11 '17 at 10:56
These combinations are fairly common. For sums < 2000 I get 13544 combinations. — PM 2Ring, Nov 11 '17 at 11:29

xxxxxxxxx · Accepted Answer · 2021-07-23T07:33:29.327

6

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};`

edited Jul 23 '21 at 07:33

answered Nov 11 '17 at 05:48

xxxxxxxxx

Thanks for the quick reply! Now that I know it's possible... is there one where all the numbers are the same? I imagine I'm asking because you have the mathematical engine and I don't... – Tony Nov 11 '17 at 05:53
Is there a way for me to learn the computational engine to find these values so I can avoid writing up questions? – Tony Nov 11 '17 at 06:05

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