There is a brief notebook that outlines the concept available in the "paper and notes" folder: Triangular Numbers and Squared Primes.nb.
To view the notebook you will need a full version of Mathematica or the CDF Player.
The python program squares primes and sums them together to determine if:
is equal to the series of squared primes.
b = triangular number (also the number base); //equal to: (r^2+r)/2
r = qg(b) = size of the base row of the triangular number; //qg(b) = 1/2(-1+sqrt(1+8b)
z = row in the triangular number; //ex. tf(10,4)=0123; tf(10,3)=456; tf(10,2)=78, etc.)
Where tf() is defined to be:
There is an interesting relationship when {b=10, r=4} where the sum of the rows in base-10, 0123 + 456 + 78 + 9, happens to work out to be the sum of the first seven squared primes.
stf(10) = 2² + 3² + 5² + 7² + 11² + 13² + 17² = 666
What I find fascinating about this relationship is the resultant value 666 is a triangular number itself. So the question then is if we were able to sum the rows of a 666 element triangle with 36 rows in base-666 would the result also be the sum of squared or cubed primes?
This program attempts to provide an answer. The base-10 number from stf(666) is massively large unfortunately at 98 digits:
37005443752611483714216385166550857181329086284892731078593232926279977894581784762614450464857290
So I'll probably have to adapt it at some point to work with CUDA or OpenCL to see if I can speed up the computations. I have a large series of primes precomputed for people to speed up the operation.