Talk:Formulas for generating Pythagorean triples: Difference between revisions

Revision as of 17:05, 8 September 2011

Revision as of 17:07, 8 September 2011

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There are a number of simple, factual, mathematical errors in the statement “'''Non-primitive triples are (by definition) solutions to both equations'''”, and the comments made above are based on these errors.

The equation (1) Dickson refers to is - <math>x = 2mn</math>, <math>y = m^2 - n^2</math>, <math>z = m^2 + n^2</math> which, together with its numerous derived methods for generating triples, will only produce a relatively few non-primitive triples when <math>xyzx y z</math> are multiplied by a square or half-square integer value. (38 possible non-primitives from the first 500 multiplying numbers, and reducing. A simple test is to try to find the triples 9.12.15, 15.20.25.etc.) The few calculable non-primitives can be given as examples, but these methods cannot provide more solutions, “by definition” or otherwise. This basic mathematical misconception well proves that examples are not proofs, and is another reason why number theorists never use them when this may infer an un-proved and un-sourced generality.

Therefore, a proof and a source are needed for the guesswork in VI. The production of all non-primitives from all the even series, without multiplication, is an important theorem.

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Subject to any further, early (hopefully mathematical) comments, I will arrange a new entry in the near future.

[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 14:32, 3 September 2011 (UTC)hoarwithy

:: Euclid and Dickson have proved that their equations produce ALL Pythagorean triples. The proofs are easy enough to follow and you can find them with little effort if you really want to understand them. However, it appears you have some other agenda. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/184.153.109.223|184.153.109.223]] ([[User talk:184.153.109.223|talk]]) 06:09, 5 September 2011 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

If you are referring to Euclid’s equation in the write-up in Entries I and II, “Pythagorean Triple”, this merely suggests non-primitives can be obtained by separately multiplying a b c, in primitives <math>a^2 + b^2 = c^2</math>, by a multiplier <math>k</math>. This was referred to in my original XIII entry as the only way non-primitives could be produced from the standard equation results, is also not appropriate to this equation because m and n produce some non-primitives, is not mentioned by Dickson or the other Professors, and is not referenced to Euclid.

I have not found any proof that Dickson’s equations produce non-primitive triples.