Talk:Pythagorean triple/Archive 1

Limitation to integers?

Why is this limited to integers? What about Pythagorean triples such as 1:1:sqrt(2) or 1:sqrt(3):2? 62.0.181.212 20:09, 4 July 2002 (UTC)

Probably because Pythagoras and his followers didn't like irrational numbers. Vicki Rosenzweig 20:17, 4 July 2002 (UTC)

They didn't even have anything beyond the natural set of numbers. Even the concept of unity and zero were shrowded in mystery. The Pythagorean thoerem itself was 'likely' discoved using arrangements of small stones in L or square shaped patterns called "gnomons". Gregory S. Hayes 22:05, 7 December 2005 (UTC), 216.196.244.34

Okay, but nowadays no one has a problem with irrational numbers. Is this article about pythagorean triples in history or the concept of pythagorean triples in modern mathematics? 62.0.183.181 09:09, 5 July 2002 (UTC)

It's about the concept of Pythagorean triples, whose definition has never changed. These integral triples are much more interesting than the general concept of triples of real numbers (x,y,z) with x²+y²=z²; pretty much all you can say about the latter is that they form an infinite cone in 3-dimensional space. AxelBoldt 10:12, 5 July 2002 (UTC)

Correct. The remarks above about irrational numbers and 1:1:sqrt(2) and the like are naive. The equation x²+y²=z² is that of a conic section in the real projective plane in homogeneous coordinates, and individual points on that curve are not of interest unless it's because they are rational points. 131.183.81.100 21:37, 7 November 2002 (UTC)

I disagree. The triples for 45-45-90 and 30-60-90 seem important/unique enough to list. —Josh Lee 02:51, Apr 7, 2005 (UTC)

Guys, we're writing an encyclopedia here. Just find an authoratative reference that indicates pythagorean triples don't need to be integers, and we can include that info here. Otherwise it stays the way it is. --Doradus 03:42, July 23, 2005 (UTC)

The whole "Pythagorean Triples as Integers" issue:

The page (incorrectly) says "The [Pythagorean theorem] states that any right triangle with integer side lengths yields a Pythagorean triple". I think that's where most of the confusion is coming from: Pythagorean triples are INTEGER solutions to the Pythagorean thoerem, not the ONLY solutions. Every other math site I've seen gets this correct: so I went ahead and corrected the entry. The definition of Pythagorean triples isn't the problem: it's the claim that all right triangles have integer sides. 24.165.161.220 14:59, 7 August 2005 (UTC), Andrew Gibson

Formula for generating Pythagorean triples

I learned that there is a formula 2p, p2-1, p2+1 for generating Pythagorean triples but it is apparently weird. What is wrong with 3, 4, 5 and 5, 12, 13? Integer 5 takes the position of the lowest and the highest integer in these two examples. How can my formula explain this? 213.226.138.241 21:18, 25 Dec 2004 (UTC)

I presume you mean $2p$, $p^{2}-1$, and $p^{2}+1$. $p=2$ gives the 3,4,5 sequence, while there is no value of $p$ that gives 5,12,13 because the latter has no two elements that differ by 2. (Nobody ever claimed your formula gave all pythagorean triples, did they?) --Doradus 16:31, July 10, 2005 (UTC)

Other methods for generating Pythagorean triples

I learned from one of my school math teachers some other ways to generate pythagorean triples:

Given the integers n and x,
We have the expressions $2x^{2}$, $2nx$ and $n^{2}$
The first number of the triple will be $2x^{2}+2nx$
The second one will be $2nx+n^{2}$
The third one, $2x^{2}+2nx+n^{2}$

Example: Being $n=3$ and $x=5$, we get:
1st number is $80$
2nd one is $39$
3rd one, $89$
Verifying: $39^{2}+80^{2}=89^{2}$

Another method: Given an integer n, the triple can be generated by the following two procedures:

First number is $2n+1$
Second number is $2n(n+1)$
Third one is $2n(n+1)+1$
Example: if $n=2$ we will obtain the triple 5, 12 and 13

First number is $4(n+1)$
Second one is $4(n+1)^{2}-1$
Third one is $4(n+1)^{2}+1$
Example: if $n=5$, we generate the triple 24, 143 and 145

I apologize for my poor editing. 168.243.25.142 05:04, 21 July 2005 (UTC), Marlino 05:06, 21 July 2005 (UTC)

Grim here, and I actually know what you're on about here, and i've edited it to be simpler.

This one ^ is for triangles for which a^2+b^2=c^2, all sides' lengths are positive intigers, and a is odd, eg:

n	a	b	c
1	(2x1)+1=3	(2x1^2)+(2x1)=4	(2x1^2)+(2x1)+1=5
2	(2x2)+1=5	(2x2^2)+(2x2)=12	(2x2^2)+(2x2)+1=13
3	(2x3^2)+1=7	(2x3^2)+(2x3)=24	(2x3^2)+(2x3)+1=25

as you can see, c is always b+1 in this 'family' of triples, and b+c=a^2, if i'm not mistaken.

there are terms of n for the perimter & area, but i cannot put them down fow the moment, as i'm supposed to be doing this for maths coursework...

i hate coursework. sorry about editing in the middle of your idea, but it was easier than copying the whole thing out again. -Grim- 06:57, 20 March 2007 (UTC)

Cleanup

This article started to look like an incoherent collection of random factoids, so I decided to clean it up. I moved the material here in case anyone wants to take care of these ugly ducklings.

Deleted this trivial observation (too trivial to waste any markup on, apparently):

if m1*n1=m2*n2, then the calculated triples will have an equal number.

and a "good idea" that remains unused:

A good starting point for exploring Pythagorean triples is to recast the original equation in the form:

b² = (c − a)(c + a)

and a random observation. Many similar patterns exist but aren't mentioned either:

It is interesting to note that there are more than one primitive Pythagorean triple with the same lowest integer, the first example is for 20, which is the lowest integer of two primitive triples: 20 21 29 and 20 99 101.

and a further example designed to impress, rather than to explain:

By contrast the number 1229779565176982820 is the lowest integer in exactly 15386 primitive triples, the smallest and largest triples it is part of are:

1229779565176982820

1230126649417435981

1739416382736996181

and

1229779565176982820

378089444731722233953867379643788099

378089444731722233953867379643788101.

For the curious, consider the prime factorisation

1229779565176982820 = 2² × 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 47.

The number of prime factors is related to the large number of primitive Pythagorean triples. Note that there are larger integers that are the lowest integer in an even greater number of primitive Pythagorean triples.

and a statement that's true but off-topic:

Fermat's last theorem states that non-trivial triples analogous to Pythagorean triples but with exponents higher than 2 don't exist.

—Herbee 22:59, 4 September 2005 (UTC)