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[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 13:35, 8 September 2011 (UTC)
:::: '''Your "simple test" is indeed simple. [9, 12, 15] is easily produced using Dickson's equations, as the example in VI ''clearly'' shows ( ''r'' = 6, ''s'' = 3, ''t'' = 6). You keep saying that you can find no proof that "Dickson’s equations produce non-primitive triples". This single example should be proof enough! Another example is triple [15, 20, 25] which is also easily produced using (''r'' = 10, ''s'' = 5, ''t'' = 10). Nowhere in the source you cite does Dickson limit himself to the primitives or non-primitives as you have repeatedly and inaccurately claimed. This is because (as he well knew) his equations apply to both cases! Another basic fact you seem to have overlooked is that Dickson's equations require that ''r'' be even. Obviously then, the "even square (or half-square) integers" you mention are also even, and are fully accounted for by his equations. To get ALL and only the non-primitives using Dickson's equations we need only find the factor pairs (''s'' and ''t'')
::::'''We can get triples [9, 12, 15] and [15, 20, 25] just as easily from Euclid using ''m'' = 2, ''n'' = 1, ''k'' = 3 for the first one , and ''m'' = 2, ''n'' = 1, ''k'' = 4 for the second. This is clearly explained and sourced in the Wikipedia article on [[Pythagorean triple|Pythagorean triples]], along with the need for parameter ''k'' when generating | |||