Talk:Formulas for generating Pythagorean triples: Difference between revisions

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'''Formulas for generating Pythagorean triples talk page.'''

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== Sum of odd numbers ==

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[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 13:35, 8 September 2011 (UTC)

:::: '''We can get triples [9, 12, 15] and [15, 20, 25] justcan asbe easily obtained from Euclid using ''m'' = 2, ''n'' = 1, ''k'' = 3 for the first one , and ''m'' = 2, ''n'' = 1, ''k'' = 45 for the second. This is clearly explained and sourced in the Wikipedia article on [[Pythagorean triple|Pythagorean triples]], along with an explanation for the the need forto introduce parameter ''k'' when generating ALL triples as opposed to just the primitives. AsBut as Euclid well knew, it is enough to consider only the set of ''primitive Pythagorean'' triples in(all of which ''a''are andgenerated ''b''by arethe coprimeequation you cite), since allALL ''non-primitive'' solutions can be generated trivially from the primitive ones. In addition to all of the primitives, the version of the equation using only parameters ''m'' and ''n'' produces an infinite number of non-primitive triples of the form [''ak'',''bk'',''ck''] where ''k > 1'' is a square or twice a square. To get only the primitives, ''m'' and ''n'' must be coprime, with ''m'' > ''n'', and one of ''m'',''n'' must be even.'''

:::: '''Your "simple test" is indeed simple. [9, 12, 15] is easily produced using Dickson's equations, as the example in VI ''clearly'' shows ( ''r'' = 6, ''s'' = 3, ''t'' = 6). You keep saying that you can find no proof that "Dickson’s equations produce non-primitive triples". This single example should be proof enough! Another example is triple [15, 20, 25] which is also easily produced using (''r'' = 10, ''s'' = 5, ''t'' = 10). Nowhere in the source you cite does Dickson limit himself to the primitives or non-primitives as you have repeatedly and inaccurately claimed. This is because (as he well knew) his equations apply to both cases! Another basic fact you seem to have overlooked is that Dickson's equations require that ''r'' be even. Obviously then, the "even square (or half-square) integers" you mention are also even, and are fully accounted for by his equations. To get ALL and only the non-primitives using Dickson's equations we need only find the factor pairs (''s'' and ''t'') which are not coprime, beginning with ''r'' = 2. A source for Dickson's proof? For starters you can look at his own footnote (34) on page 165 of the book you have cited, but not read very carefully.'''

:::: '''Your "simple test" is indeed simple. [9, 12, 15] is just as easily produced using Dickson's equations, as the example in VI ''clearly'' shows ( ''r'' = 6, ''s'' = 3, ''t'' = 6). You keep saying that you can find no proof that "Dickson’s equations produce non-primitive triples". This single example should be proof enough! AnotherYour other example is, triple [15, 20, 25] which is also easily produced using (''r'' = 10, ''s'' = 5, ''t'' = 10). Nowhere in the source you cite does Dickson limit himself to the primitives (or non-primitives) as you have repeatedly and inaccurately claimed. This is because (as he well knew) his equations apply to both cases! Another basic fact you seem to have overlooked is that Dickson's equations require that ''r'' be even. Obviously then, the "even square (or half-square) integers" you mention are also even, and are fully accounted for by his equations. To get ALLas and only themany non-primitives using Dickson's equations as we want, we need only findrequire thethat factor pairs ( ''s'' and ''t'' ) whichshare area notcommon coprimefactor, beginningand begin with ''r'' = 2. A source for Dickson's proof? For starters you can look at his own footnote (34) on page 165169 of the book you have cited. See also P.G. Egidi, butD. notGambioli, readA. veryBottari, carefullyand H. Shotten [''Footnotes 35, 36, 39, and 36a on the same page'']. '''

::::'''I assume you agree that Dickson's equations produce all the primitives. This is true only when ''s'' and ''t'' are coprime as in Line 1 below. (If they were not coprime, then we could remove common factor ''k'' as shown on Line 2. (Clearly, if ''k'' is a factor of ''s'' and ''t'', it is also a factor of ''r''). But if instead we apply common factor ''k'' to our primitive triple(s) [ ''x, y, z''], we get the non-primitive triple(s) ['' x', y', z'''] as shown on Line 3 where ''r', s', t' '' share common factor ''k'' > 1. '''

:::::<math>\begin{array}{*{35}{l}}

1) & x=\text{ }(r+s),\text{ }y=(r+t),\text{ }z=(r+s+t) \\

2) & {x}'=(r+s)k,\text{ }{y}'=(r+t)k,\text{ }{z}'=(r+s+t)k \\

3) & {x}'=({r}'+{s}'),\text{ }{y}'=({r}'+{t}'),\text{ }{z}'=({r}'+{s}'+{t}') \\

\end{array}</math>

::::'''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)

:::: '''I will not discuss this further. However, be advised that your webpage does not meet Wikipedia's standards as a source or reference. Adding it to these pages will be considered vandalism.''' <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]]) 17:02, 3 October 2011 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

How strange you no longer wish to discuss this. In much the same way you took it upon yourself to delete a section without discussion, and when asked for justification have been unable to come up with any despite claiming it is readily available. This surely suggests it is you who are guilty of vandalism and are hardly in a position to accuse others of it.

[[User:Hoarwithy|Hoarwithy]] ([[User talk:Hoarwithy|talk]]) 22:52, 3 October 2011 (UTC)

== Combine IV and V? ==

Suggestion: Sections IV and V might be combined into a section called "Variations on Euclid"? <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]]) 06:06, 1 October 2011 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== weird TeX ==

Someone who worked here extensively wrote a lot of stuff like this:

: {{a}^{2}} + {{b}^{2}} = {{c}^{2}}

within math tags, instead of

: a^2 + b^2 = c^2

and did a bunch of other things that look as if they were done by a lunatic or at least someone not particularly familiar with [[TeX]]. Sometimes curly braces are necessary (e.g. if you have more than one character in a subscript or a superscript, such as a_{2n}. But sometimes they're at best clutter. Is there some reason why it was done that way? [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 23:11, 11 October 2011 (UTC)

:Heh heh, I can't claim credit for most of them, but in one edit in May 2010 I did put in a couple of instances of (1^{2}+2^{2}) etc., which were subsequently augmented by someone else to ({{1}^{2}}+{{2}^{2}}) etc. When you simplified the others you missed mine, so I've done them now. Don't worry about me -- I've recently upped my dosage of lunatic meds! [[User:Duoduoduo|Duoduoduo]] ([[User talk:Duoduoduo|talk]]) 19:43, 19 October 2011 (UTC)

==Deleting my contribution==

I made a contribution to this article little more than a week ago on "the universal set of Pythagorean triples." Someone removed it. Why? I made reference to an article that contains rigorous proof of a matrix of Pythagorean triples that form the universal set of triples. This is the ultimate in finding the triples. Why was it removed, and what must I do right this time to have it maintained? Should I rather write my own article, independent on this one, for finding any Pythagorean triple? <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/119.46.251.60|119.46.251.60]] ([[User talk:119.46.251.60|talk]]) 07:59, 6 December 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

: Please see the page [[WP:OR]]. The short summary is that Wikipedia does not publish original research, and that's what you were adding to the page. (Also, for what it's worth, your method is a relatively minor variation of some of the formulas on this page; for example, if we set ''x'' = ''m''/''n'' and rescale then your triple is the same as the "usual" triple from [[Pythagorean_triple#Generating_a_triple|Euclid's formula]].) --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 19:35, 6 December 2013 (UTC)

== Generating triples when one side is known ==

This section is wrong so someone should fix it:) Euclid's formula is designed to generate only primitive triples, but the text claims that it will generate all of them. Omitting the restriction that m and n should be coprime allows some non-primitive triples to be generated, but not all of them. E.g. in the example given with b = 24, (18, 24, 30) and various others are missed. In order to generate all possible triples, the factor k as in http://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple needs to be included. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/97.77.50.20|97.77.50.20]] ([[User talk:97.77.50.20|talk]]) 04:30, 7 February 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Article impossible to understand if you don't know the topic already ==

The properties of this progression are:

(a) the whole numbers are those of the common series and have unity as their common difference; (b) the numerators of the fractions, annexed to the whole numbers, are also the natural numbers; (c) the denominators of the fractions are the odd numbers, <math>3,\text{ }5,\text{ }7,\text{ }9,</math> etc.

* What is a [[common series]]?

* What is [[unity]]?

* What is a [[common difference]]?

* What does [[annexing]] mean?

[[Special:Contributions/91.83.139.187|91.83.139.187]] ([[User talk:91.83.139.187|talk]]) 07:40, 22 January 2019 (UTC)

:{{Fixed}} —&nbsp;[[User:UnladenSwallow|UnladenSwallow]] ([[User talk:UnladenSwallow|talk]]) 19:30, 6 November 2021 (UTC)

::::'''We can get triples [9, 12, 15] and [15, 20, 25] just as easily from Euclid using ''m'' = 2, ''n'' = 1, ''k'' = 3 for the first one , and ''m'' = 2, ''n'' = 1, ''k'' = 4 for the second. This is clearly explained and sourced in the Wikipedia article on [[Pythagorean triple|Pythagorean triples]], along with the need for parameter ''k'' when generating ALL triples. As Euclid well knew, it is enough to consider only the set of primitive Pythagorean triples in which ''a'' and ''b'' are coprime, since all non-primitive solutions can be generated trivially from the primitive ones.'''