Although it appears that the really good solutions typically make use of the factorization,A Pythagorean triplet is a set of three natural numbers, $a<b<c$, for which,

$$a^2+b^2=c^2$$

For example, $3^2+4^2=9+16=25=5^2$.

There exists exactly one Pythagorean triplet for which $a+b+c=1000$.

Find the product $abc$.

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I derived (in the forums) the factorization

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Though it presents no noticeable speed increase over the $mn$-factorization with a case as small as 1000, my solution appears to be more efficient in the specific case of identifying Pythagorean triplets by their sum. Seeing as this idea hadn't been presented previously in the forums or in the solution document, I was wondering if the solution documents are ever updated with new and improved solutions and if yes, would my above solution qualify?