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== Sections XIII and V - proposal for reinstatement ==
Professor Dickson, on Page 169 of his “History of the Theory of Numbers” Vol.II. Diophantine Analysis, Carnegie Institution of Washington, Publication No. 256, 12+803pp read online at University of Toronto here[http://www.archive.org/stream/historyoftheoryo02dickuoft#page/168/mode/2up] 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) (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), and Professor Paulo Ribenboim in “Fermat’s Last Theorem for Amateurs” (Pages 7 and 8) - (New York: Springer-Verlag, 1999) 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.
::'''“equivalent to (1)” refers to the chapter heading "Methods of Solving <math>{{x}^{2}}+{{y}^{2}}={{z}^{2}}</math> in Integers", so it appears you have either misread or misunderstood Dickson. Non-primitive triples are (by definition) solutions to this equation.'''
So all triples were not seen by Dickson and other number theory authors to be calculable by his simple equations.
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It appears that the combination of Entries V and XIII under V is unsatisfactory They have not been fully combined, or give essential information about proof, source or correct mathematical context, and further changes are necessary in the near future.
[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 11:29, 22 August 2011 (UTC)hoarwithy
::'''Dickson’s method clearly allows for the non-primitives as the example in VI shows. In fact, when s and t have a common factor, his equations produce ONLY the non-primitives. Since Dickson’s method also produces all the primitives, we have only to use integer ''k'' as a multiple to obtain any desired non-primitive [''ak,bk,ck'']. Alternatively, we can apply ''k'' to ''r'', ''s'', or ''t''.'''
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