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### The explicit monoid law on the differences of two square integers?

$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$ is a famous identity. Now substitute $b=ib’,d=id’$ to obtain $$(a^2-b^2)(c^2-d^2)=(ac-bd)^2-(ad-bc)^2$$.
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### What is the maximum relative density of squares congruent to m modulo n for chosen m and n?

I just did a brute-force search for $m, n \le 1000$, and could not find a higher density than 8/3. There are ties with n = 96, 216, 384, 600, and 864, though.
• 8,523

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