Talk:Pythagorean triple

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated B-class, Mid-priority)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
Mid Priority
 Field: Number theory
One of the 500 most frequently viewed mathematics articles.

A New Property -- Can Anyone Find it in the Literature?[edit]

For primitive PT's: c2+2ab and c2-2ab are both odd squares.

The proof is simple: c2+2ab=a2+b2+2ab=(a+b)2. The latter is an odd square because exactly one of a, b is odd. Similarly for c2-2ab.

I don't see this in my copy of Sierpinski's "Pythagorean Triangles"... has anybody seen it somewhere? If so, is it worth adding to "Elementary Properties..."? (And if not, this has to be the most trivial "original research" you can imagine.) CirclePi314 (talk) 09:20, 12 February 2015 (UTC)

This article tends to be a magnet for things that match the description in your final parenthetical. --JBL (talk) 16:29, 12 February 2015 (UTC)

Dubious passage on Platonic sequences[edit]

The following was put into the article's section on Platonic sequences on 28 June 2007 by someone who is no longer on Wikipedia:

It can be shown that all Pythagorean triples are derivatives of the basic Platonic sequence (xyz) = p, (p2 − 1)/2 and (p2 + 1)/2 by allowing p to take non-integer rational values. If p is replaced with the rational fraction m/n in the sequence, the 'standard' triple generator 2mn, m2 n2 and m2 + n2 results. It follows that every triple has a corresponding rational p value which can be used to generate a similar triangle (one with the same three angles and with rational sides in the same proportions as the original). For example, the Platonic equivalent of (6, 8, 10) is (3/2; 2, 5/2). The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.

This seems wrong to me. First, as far as I can see, the Platonic sequence p, (p2 − 1)/2 and (p2 + 1)/2 does not give rise to (3/2, 2, 5/2). Second, as far as I can see it is impossible to find any rational p for which this formula gives something that scales to (6, 8, 10). Third, I think the same is true for the different Platonic sequence formula given in the article Diophantus II.VIII.

I conclude from this that in fact the Platonic sequence can generate all primitive triples but not all triples. This feeling is reinforced by the observation in the above passage that the Platonic sequence can be used to derive the 'standard' formula, which generates all primitive triples but which does not generate all triples.

Without objection, I'm going to replace the above passage with the following:

''It can be shown that all primitive Pythagorean triples can be derived from the basic Platonic sequence (xyz) = p, (p2 − 1)/2 and (p2 + 1)/2 by allowing p to take non-integer rational values. If p is replaced with the rational fraction m/n in the sequence, the 'standard' primitive triple generator 2mn, m2 n2 and m2 + n2 results. It follows that every primitive triple has a corresponding rational p value which can be used to generate a similar triangle (one with the same three angles and with rational sides in the same proportions as the original). For example, the Platonic equivalent of (56, 33, 65) is generated by p = m/n = 7/4 as (p, p2 –1, p2+1) = (56/32, 33/32, 65/32). The Platonic sequence itself can be derived by following the steps for 'splitting the square' described in Diophantus II.VIII.

Loraof (talk) 17:58, 8 April 2015 (UTC)

I don't understand your objection. If I can make something similar to every primitive triple then I can make something similar to every triple (since every triple is similar to a primitive one). Though the example is confused, since to get the triple (3/2, 2, 5/2) one should put in p = 2 (an integer). --JBL (talk) 18:32, 8 April 2015 (UTC)
Thanks. First, I found (and still find) the phrase "triples are derivatives of ..." confusing. From your comments I can see that this was intended to mean with unrestricted scaling. Second, I was misled by the statement that the Platonic sequence, previously defined as p, (p^2-1)/2, (p^2+1)/2, is exemplified by (3/2, 2, 5/2), which says that p=3/2 and 2=((3/2)^2-1)/2, which is not true. It should be (2, 3/2, 5/2). I'm going to clarify the derivatives wording, and replace the example with one that is correctly sequenced and has a fractional p. Loraof (talk) 19:16, 8 April 2015 (UTC)
Yes, you're certainly right that it was not clearly written at all. Your edit was a big improvement; in a second I will tweak it a bit more. --JBL (talk) 21:30, 8 April 2015 (UTC)