Magical Squareception

We study restrictions on the number of prime factors a primitive 3-by-3 magic square of squares may have. When writing the middle entry as $e^2$, our result excludes both $e=p^a$ and $e=p^aq^b$ for prime $p,q$ and natural $a,b$. Consequently, the center value must have at least three distinct prime factors. We combine classical three-parameter form of a magic square of squares with Gaussian-integer factorizations from the four opposing pairs. Primitivity further constrains the possible prime factors among the deviations. Furthermore, in the two-prime case, we show that their linear relations grant bounds on auxiliary integers parameters which are completely independent of the prime exponents. We further this observation with congruence and norm equivalencies to then eliminate the remaining cases before finishing by discussing how the method does not extend to centers with three or more distinct prime factors.