Talk:Formulas for generating Pythagorean triples: Difference between revisions

Revision as of 06:07, 1 October 2011

Revision as of 21:27, 1 October 2011

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::::'''Thus, given an arbitrary primitive [ ''x, y, z''] with ''s'','' t'' coprime, we get all of its ''non-primitive'' multiples too. The latter have the form ['' x', y', z' '' ] where ''r', s', t''' share common factor ''k'' > 1.'''

I will clarify the position -

In July you arbitrarily deleted my entry XIII, which showed both source and proof that all Pythagorean triples, including non-primitives, can be calculated uniquely from the factors of each and every even number squared. This included a new and necessary algebraic extension to an equation first published by Professor Dickson. You substituted as a joint site your own anonymous, un-proved entry, with its single example, now VI.

You claimed there is an earlier proof to mine, which “can be found with little effort” saying, above, that Dickson’s own footnote (34) refers to his proof that his equation produces all non-primitives. '''This is incorrect'''. In his Amer. Math. Monthly, 1, 1894, 8, article he specifically confines himself to primitives. For the other references you quote, when providing them Dickson says these give the same rule as his.

Dickson says his equation is equivalent to Euclid’s. You had earlier claimed Euclid’s equation produces all the non-primitives, but '''this was also incorrect''' because, in your first paragraph above, you now agree a separate multiplier (k) needs to be applied to the x y z (varied to a b c) of his produced primitives. (In your opening sentence you appear to be suggesting k can be applied to m and n, but this would also be incorrect.)

'''I pointed out''' that Euclid’s equation does produce a tiny minority of non-primitives, being a classic case that, in mathematics, examples are never taken as proofs . Yet, in your second paragraph above you repeat your assertion that a “single example should be proof enough”, which is '''mathematically incorrect'''.

'''You explain and give equations 1 – 3 above to show Dickson’s xyz and str can be multiplied by k to give non-primitives.''' Your earlier footnote to the above, removed recently, appeared to claim first publication for yourself. Yet Section F, point 7 of my website (http://www.calculatingpythagoreantriples.org.uk) referred to in my entry XIII, which you deleted has as its final conclusion “X Y Z A B G are equally affected by multiplication and division”. The A B G in my equations being the s t r in Dickson’s equations.

You have had two months to demonstrate in discussion an earlier proof to mine that all triples can be calculated from Dickson’s equations, and that they can be calculated from the factors of the even numbers <math>r^2/2</math>. You have failed to do so, just as I, and others, had failed to find one. I propose shortly replacing the combined entry VI with another.

May I suggest that deleting or changing another mathematicians website without discussion is, at best, vandalism. It would have at least been courteous to raise your comments in discussion before arbitrary, anonymous censorship.

[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 21:27, 1 October 2011 (UTC)

== Combine IV and V? ==