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A New Property -- Can Anyone Find it in the Literature?
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)
Another Property -- Can Anyone Find it in the Literature?
Some triplet comes also from:
- Well, we know that , so if and only if . For some Pythagorean triplets it is the case that , but there are also many Pythagorean triplets for which . 𝕃eegrc (talk) 14:13, 19 May 2016 (UTC)
Dubious passage on Platonic sequences
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 (x, y, z) = 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 (x, y, z) = 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.
- 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)
Since for Pythagorean primitive triples (a,b,c) we have gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. I propose to give this as a property in the article. — Preceding unsigned comment added by 126.96.36.199 (talk) 12:43, 6 October 2015 (UTC)
Rightarrow versus implies
In Google Chrome on my laptop running Windows Vista, in the Points on a unit circle section, the \implies produces an arrow which is broken in the middle, but the \Rightarrow looks fine. This may be why the IP 188.8.131.52 changed it. — Anita5192 (talk) 21:14, 29 January 2016 (UTC)
- Would it be better just to replace the arrow with the words "and so" or similar? --JBL (talk) 21:20, 29 January 2016 (UTC)
- My general preference in situations like this is to spell it out in words. I think doing it that way makes it less WP:TECHNICAL. —David Eppstein (talk) 23:19, 29 January 2016 (UTC)
It seems reasonable to add a link to https://en.wikipedia.org/wiki/Boolean_Pythagorean_triples_problem under "See also".
Since I am an author of the article referenced in that page, I don't know whether I should add the link just myself (which seems harmless, since it is another Wikipedia page, and obviously it is related to the current page). Oliver Kullmann (talk) 16:34, 5 June 2016 (UTC)