Talk:Formulas for generating Pythagorean triples: Difference between revisions

Revision as of 06:12, 13 September 2011

Revision as of 20:14, 13 September 2011

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[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 13:35, 8 September 2011 (UTC)

:::: '''[9, 12, 15] and [15, 20, 25] can be easily obtained from Euclid using ''m'' = 2, ''n'' = 1, ''k'' = 3 for the first one , and ''m'' = 2, ''n'' = 1, ''k'' = 5 for the second. This is clearly explained and sourced in the Wikipedia article on [[Pythagorean triple|Pythagorean triples]], along with an explanation for the the need to introduce parameter ''k'' when generating ALL triples as opposed to just the primitives. But as Euclid well knew, it is enough to consider only the set of ''primitive'' triples (all of which are generated by the equation you cite), since ALL ''non-primitive'' solutions can be generated trivially from the primitive ones. In addition to all of the primitives, the version of the equation using only parameters ''m'' and ''n'' produces an infinite number of non-primitive triples of the form [''ak'',''bk'',''ck''] where ''k > 1'' is an "evena square (or half-square)twice integera square". To get only the primitives, ''m'' and ''n'' must be coprime, with ''m'' > ''n'', and one of ''m'',''n'' must be even.'''