Using the definition of the imaginary number (i) as a set of simultaneous real numbers, we are able to offer novel insights to the manipulation of the complex space using only real numbers. We can visualize how trial division / Fermat's method operates on this space and increase efficiency by combining methods.
The simultaneous real numbers take the form [(real - imaginary),(real + imaginary)]. Some examples include:
- i → [-1,1]
- 2i → [-2,2]
- -i → [1,-1]
- (8+5i) → [3,13]
- (-2-3i) → [1,-5]
The papers provide details on how the complex space maps to real numbers, how all complex number operations are carried out with only real numbers, and factor properties.
Please see the Prime-Factorization branch for factorization routines based on these insights.