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I'm searching a trusted historical sources about primitive Pythagoras triplets as being powerful integers (numerical examples), or a notable work of impossibility of such a triples, but couldn't find despite asking this question here at MSE and MOF

I do expect some old references for this issue, as when and who stated this issue, and what are the consequences of this problem on the important unsolved problems in mathematics

Please if any information available, provide exact dates with references.

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    $\begingroup$ *I do expect some old references for this issue*—depending on what is considered 'old', of course, but it seems to me that powerful numbers weren't considered before the 1930s...? By the way, here is the (OPs) MSE-question asking whether primitive Pythagorean triples consisting of powerful numbers only can exist. $\endgroup$ – Ben Jun 24 '16 at 15:38
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    $\begingroup$ Pythagorean triples are old in history even before Pythagoras, I think the oldest are during Babelyons old civilizations (Iraq nowadays), but we don't know for sure wither all their properties are discovered, if the suggested (most likely new property is established), then so much bless to mathematics, and if it is refuted (by a single counter example), then nothing would be lost at all, it seems to me that we are in need of the help of supercomputer first, then we can talk about it more seriously ! $\endgroup$ – Sophyan Gharz Jun 24 '16 at 21:25
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    $\begingroup$ In fact, Pythagorean triples are under consideration for a very long time, but (to my knowledge) powerful numbers weren't. $\endgroup$ – Ben Jun 24 '16 at 23:59
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    $\begingroup$ Actually, if we assume this is an established fact, then few unsolved problems would be an immediate results, but the proof of this conjecture also seems so simple as claimed or may be so difficult as unsolved problems, let me arrange one proof once I complete it soon, the conjecture itself is very interesting since it should had been noted long ago, (but we hope it fails completely), the interesting result would be so surprising since it would prove Beal's conjecture as well, powerful triplet is indeed so powerful approach to many unsolved problems I think! $\endgroup$ – Sophyan Gharz Jun 25 '16 at 1:16
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    $\begingroup$ @SophyanGharz Thanks for you to activate this unsolved problem, which is relevant to the most famous fact in mathematics, that is Pythagoras theorem, however it would be very nice of you if you can prove it, but be aware that there are better places for such proof than here, I guess also many secretly had known the simple trick for this puzzle, so let them enjoy secretly the discovering sense of great puzzles for a while, actually soon there would be so many of them hunting the secret at nearly the same time, and of course the remaining parts are not so interesting, but would be hurting badly $\endgroup$ – bassam karzeddin Jun 30 '16 at 8:32

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