Talk:Formulas for generating Pythagorean triples: Difference between revisions

Revision as of 11:33, 22 August 2011

Revision as of 22:37, 23 August 2011

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== Sections XIII and V - proposal for reinstatement ==

Professor Dickson, on Page 169 of his “History of the Theory of Numbers” <ref>Dickson, Prof L. E., (1920), History of the Theory of Numbers, Vol.II. Diophantine Analysis, Carnegie Institution of Washington, Publication No. 256, 12+803pp Readread online -at University of Toronto: here[http://www.archive.org/stream/historyoftheoryo02dickuoft#page/168/mode/2up]</ref> makes only one comment, that his solution is “equivalent to (1)”. (1) appears at the foot of Page 165 as the standard two squares method, which is universally recognised as producing only primitive triples. Dickson does not refer to non-primitives. Professor Loomis in his book “The Pythagorean Proposition” (Pages 19 and 21) <ref>(Loomis, E. S. The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 1968</ref>), and Professor Paulo Ribenboim in “Fermat’s Last Theorem for Amateurs” (Pages 7 and 8) <ref>Ribenboim, P. Fermat's Last Theorem for Amateurs.- (New York: Springer-Verlag, 1999.</ref>) discuss Dickson’s solution and refer only to primitives and not to non-primitives, while the latter says “Hence necessarily both u and v′ (providing Dickson’s s and t) are squares” thereby saying necessarily only primitives can arise.

So all triples were not seen by Dickson and other number theory authors to be calculable by his simple equations.

In mathematics, examples are usually considered to be insufficient to prove an important theorem about the production of all non-primitives. (Otherwise Fermat’s Last Theorem would easily be “provable”!). It is probable the Professors may have known that stating the universal generality of non-primitive production would need a proof. A publication to inform readers would surely need to refer to one, either from published literature, or be provided. Otherwise it is mere speculation. A proof was provided in the deleted Entry XIII, and is summarised in www.calculatingpythagoreantriples.org.uk<ref>[http://www.calculatingpythagoreantriples.org.uk]</ref>

As Dickson did not include non-primitives in his results, the authorship or source of so fundamental a theorem should be provided. The above website was first published in the year 2000 and provides an original source.