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After running into this terminology on the "Formulas for generating Pythagorean triples" Wikipedia page, I was curious whether all triples fit into these categories.

The article states:

Plato: c - b = 1, Pythagoras: c - a = 2, Fermat: |a - b| = 1

Another page lists these as the first primitive triples where c<100:

(3, 4, 5)       (5, 12, 13)     (8, 15, 17)     (7, 24, 25)
(20, 21, 29)    (12, 35, 37)    (9, 40, 41)     (28, 45, 53)
(11, 60, 61)    (16, 63, 65)    (33, 56, 65)    (48, 55, 73)
(13, 84, 85)    (36, 77, 85)    (39, 80, 89)    (65, 72, 97)

However, I noticed that some sets like the (36, 77, 85) triple have a very large difference between each integer. Does this mean that not all pythagorean triples are part of one of those families? And if so, is there a formula for generating all the pythagorean triples?

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General solution. Let $m\gt n$ be any pair of integers. To get P. triples, let $a=m^2-n^2,\ b=2mn,\ c=m^2+n^2$

https://en.wikipedia.org/wiki/Pythagorean_triple

Above is a complete discussion.

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  • $\begingroup$ Does this mean that not all Pythagorean triples are part of one of those families? How about the triple [3,4,5], it satisfies all of the requirements (Plato: c - b = 1, Pythagoras: c - a = 2, Fermat: |a - b| = 1) so does that mean that it is part of all of them or none of the families? $\endgroup$ – Cobie Fisher Sep 30 '18 at 7:52
  • $\begingroup$ The formula (Euclid's) gives all triplets. As far as belonging to specific families, it is obvious from the list you presented that some do not. I don't understand what you are looking for. $\endgroup$ – herb steinberg Sep 30 '18 at 22:35
  • $\begingroup$ I don't understand what is troubling you. Some triplets don't belong to any of the families, some belong to one or more. Looking at your table, it is obvious. (3,4,5) belongs to all three. $\endgroup$ – herb steinberg Oct 1 '18 at 19:48

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