Let Us Count the Ways

I was curious as to how to count the number of ways that a given number can be represented in Loeschian form based on its prime factorisation. I knew there was such a method for counting how many ways a number can be represented as a sum of two squares. I got Gemini to investigate, compare and summarise the results. Notice the use of the ceiling function when determining the number of essentially distinct pairs. Remember also that the invalid primes become valid if raised to an even power but they don't contribute to the count.

The algebraic frameworks for expressing integers as \(x^2 + xy + y^2\) and \(x^2 + y^2\) mirror each other directly through their respective complex integer rings.

Characteristic	Loeschian Form \(x^2 + xy + y^2\)	Sum of Two Squares \(x^2 + y^2\)
Complex Ring	Eisenstein Integers \(\mathbb{Z}[e^{2\pi i/3}]\)	Gaussian Integers \(\mathbb{Z}[i]\)
Invalid Primes (Require Even Exponents)	\(p \equiv 2 \pmod 3\)	\(p \equiv 3 \pmod 4\)
Generating Primes (Exponents \(b_i\))	\(p \equiv 1 \pmod 3\)	\(p \equiv 1 \pmod 4\)
Neutral Prime	\(p = 3\)	\(p = 2\)
Unit Symmetries (Multiplier)	6	4
Total Ordered Pairs \((x,y)\)	\(6 \prod (b_i + 1)\)	\(4 \prod (b_i + 1)\)
Essentially Distinct Pairs \(\{x,y\}\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)

Factorization Conditions

Both systems demand that primes which cannot be represented natively by the quadratic form must be "squared away" with even exponents in the prime factorization. For Gaussian integers, primes congruent to \(3 \pmod 4\) (such as 3, 7, 11) cannot form sums of two squares. For Eisenstein integers, primes congruent to \(2 \pmod 3\) (such as 2, 5, 11) cannot be Loeschian. If any of these restricted primes have an odd exponent, the integer cannot be represented in that form, and the total count is zero.

Symmetry and Distinct Ways

The difference in the total number of ordered coordinate pairs stems entirely from the geometry of the complex rings. Gaussian integers form a square lattice with 4 units (\(\pm 1, \pm i\)), leading to a rotational symmetry multiplier of 4. Eisenstein integers form a hexagonal lattice with 6 units (\(\pm 1, \pm e^{2\pi i/3}, \pm e^{4\pi i/3}\)), leading to a multiplier of 6.

Once these geometric symmetries are factored out by dividing by the respective unit count, both systems calculate the number of essentially distinct, non-negative integer pairs using the exact same ceiling function formula. The generating primes dictate this final count, while the unique neutral prime in each system scales the base integer without creating new algebraic variations.