How to calculate prime elements most quickly for a hypercomplex space that can be described using the following method...
n≠0? CD[n]²=-CD[n-1]
n=0? CD[n]²=CD[n]
n>m? CD[n]*CD[m]=CD[n]
The prime elements take the form:
a*CD[0]+b*CD[1]+c*CD[2]+d*CD[3]...
where a, b, c, d... are integers.
a*CD[0]+b*CD[1], for example, are Gaussian prime elements where the following conditions hold:
a*b=0 & IsPrime[Abs[a+b]] & Abs[a+b] %4=3? return True
return IsPrime[a²+b²]
Since in the third dimension, depending on the value range, results from lower value ranges may be excluded, the question is whether, in the Python function...
def gigatime(ga, gb):
gc = [0, 0, 0]
gc[1] -= ga[2] * gb[2]
gc[2] += ga[2] * gb[1]
gc[2] += ga[2] * gb[0]
gc[2] += ga[1] * gb[2]
gc[2] += ga[0] * gb[2]
gc[0] -= ga[1] * gb[1]
gc[1] += ga[1] * gb[0]
gc[1] += ga[0] * gb[1]
gc[0] += ga[0] * gb[0]
return gc
...for values where |ga[0]|+|ga[1]|+|ga[2]|>1 and ga[0]∈ℤ, ga[1]∈ℤ, ga[2]∈ℤ,
and |gb[0]|+|gb[1]|+|gb[2]|>1 and gb[0]∈ℤ, gb[1]∈ℤ, gb[2]∈ℤ,
there are values |gc[0]|+|gc[1]|+|gc[2]|>1 and gc[0]∈ℤ, gc[1]∈ℤ, gc[2]∈ℤ that cannot be generated by the equation gc=gigatime(ga,gb).
What I have found out so far is that, among the results, most of the values [a, b, c],
a² + b⁴ + c⁸, are prime numbers.
Here are the potentially secure values for the range |a|, |b|, |c| ≤ 7:
[−7,−2,−6],[−7,−2,6],[−7,2,−6],[−7,2,6],[−6,−6,−7], [−6,−6,7],[−6,−5,0],[−6,5,0],[−6,6,−7],[−6,6,7], [−5,−2,0],[−5,2,0],[−4,−5,0],[−4,5,0],[−3,0,−2], [−3,0,2],[−2,−5,0],[−2,−1,0],[−2,1,0],[−2,5,0], [−1,−6,−2],[−1,−6,2],[−1,−4,−2],[−1,−4,2], [−1,−2,0],[−1,0,−4],[−1,0,−2],[−1,0,2], [−1,0,4],[−1,2,0],[−1,4,−2],[−1,4,2], [−1,6,−2],[−1,6,2],[0,−6,−5],[0,−6,−1], [0,−6,1],[0,−6,5],[0,−4,−5],[0,−4,−1], [0,−4,1],[0,−4,5],[0,−3,−2],[0,−3,2], [0,−2,−7],[0,−2,−5],[0,−2,−3],[0,−2,−1], [0,−2,1],[0,−2,3],[0,−2,5],[0,−2,7], [0,−1,−6],[0,−1,−2],[0,−1,−1],[0,−1,1], [0,−1,2],[0,−1,6],[0,1,−6],[0,1,−2], [0,1,−1],[0,1,1],[0,1,2],[0,1,6], [0,2,−7],[0,2,−5],[0,2,−3],[0,2,−1], [0,2,1],[0,2,3],[0,2,5],[0,2,7], [0,3,−2],[0,3,2],[0,4,−5],[0,4,−1], [0,4,1],[0,4,5],[0,6,−5],[0,6,−1], [0,6,1],[0,6,5],[1,−6,−2],[1,−6,2], [1,−4,−2],[1,−4,2],[1,−2,0],[1,0,−4], [1,0,−2],[1,0,2],[1,0,4],[1,2,0], [1,4,−2],[1,4,2],[1,6,−2],[1,6,2], [2,−5,0],[2,−1,0],[2,1,0],[2,5,0], [3,0,−2],[3,0,2],[4,−5,0],[4,5,0], [5,−2,0],[5,2,0],[6,−6,−7],[6,−6,7], [6,−5,0],[6,5,0],[6,6,−7],[6,6,7], [7,−2,−6],[7,−2,6],[7,2,−6],[7,2,6]