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Finding Solutions to a System of Diophantine Equations

Mathematics Asked by GodOrGovern on November 26, 2020

I’m trying to find triplets of integer $(x, y)$ pairs – $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ – that satisfy the following equations:

$$
{x_1}^2 + {y_1}^2 = {x_2}^2 + {y_2}^2 = {x_3}^2 + {y_3}^2 \
x_1 + x_2 + x_3 = 5 \
y_1 + y_2 + y_3 = 0 \
(x_1, y_1) neq (x_2, y_2) \
(x_1, y_1) neq (x_3, y_3) \
(x_2, y_2) neq (x_3, y_3)
$$

Currently, for each integer $c$ that can be written as the sum of two squares, I create a list of all possible integer pairs $(a, b)$ for which $a^2 + b^2 = c$ and then check all possible triplets from each list for validity (actually I only need to check all possible combinations of 2 pairs, but the idea is the same). The only constraints I’ve found are $c$ must be divisible by 5 and $a$ and $b$ must have different parity. This is pretty inefficient, as only a very small subset of possible values of $c$ produce valid triplets. Checking the validity of any given triplet is relatively easy, so I’m hoping someone can help me find further constraints on the values of $c$. It is also entirely possible that there is some other, more efficient method of finding these triplets. Either way, help would be much appreciated.

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