Content deleted Content added
Line 57:
::'''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 ''st''.'''
== However ==
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>xyz</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.
| |||