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I would like to complement you on a well written article but alas mathematically, for me at least it lacks coherency and substance. It takes one slowly through some seemingly simple yet carefully selected equations allowing triples to be generated from (1) odd numbers (2) even numbers and then gives a table with examples of both.
As the given equations do not generate all triples we are then presented with the formulas for Euclid's numbers given in The Elements (which book, which proposition because I can't find it!) but nowhere is there an example of these coherently structured numbers. Also Euclid's numbers combine to give a three term equation which is just the 'Quarter Squares Rule', The Elements, book 2, proposition 8 and which was posted in a comment by anonymous on 20/12/2012.
Incidentally, about the author says that you are from the land of Aryabhatt the Elder who way back in the 5th century knew that the partial sums of the cubes, 1^3 + ... +n^3 = a triangular number squared and from which one can easily derive the 'Quarter Squares Rule'.
Apart from a hint at their existence you do not raise the question of infinite odd and even series of Pythagorean triples which is essential if one is to make sense of a table of 10,000 triples which one can download from the internet. After I managed to sort series out to my satisfaction I programmed Excel to generate any series of triples from just two number inputs which at the start of your article was the purported aim of the ancients.
And what about simply getting ones hands dirty and saying c^2 - b^2 = integer = a^2 IF and ONLY IF the integer equals a perfect square particularly in these days of computer spreadsheets.
Finally, what's the connection if any between Pythagorean triples together with quadruples and' Fermat's Last Theorem'.