This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 |
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)
The following was put into the article's section on Platonic sequences on 28 June 2007 by someone who is no longer on Wikipedia:
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:
Loraof ( talk) 17:58, 8 April 2015 (UTC)
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 128.84.127.40 changed it. — Anita5192 ( talk) 21:14, 29 January 2016 (UTC)
Some triplet comes also from:
A=3, B=4, C=5 is the most famous. — Preceding unsigned comment added by 94.81.217.96 ( talk) 13:15, 19 May 2016 (UTC)
It seems reasonable to add a link to /info/en/?search=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)
Done D.Lazard ( talk) 07:52, 6 October 2016 (UTC)
In the text it says that "A proof of the necessity that be expressed by Euclid's formula for any primitive Pythagorean triple is as follows". However, Euclid's formula only gives primitive Pythagorean triples with even. It is for instance obvious that we cannot find m and n such that , , . As pointed out earlier, if is even and , Euclid's formula gives where is a PPT. For instance and gives .
So a few places, primitive Pythagorean triple should be replaced by primitive Pythagorean triple where is even. — Preceding unsigned comment added by HelmerAslaksen ( talk • contribs) 19:44, 6 September 2016 (UTC)
HelmerAslaksen ( talk) 00:34, 6 October 2016 (UTC)
Take a look at Pythagorean triple#Interpretation of parameters in Euclid's formula. There we see that n/m is the lowest terms representation of tan(θ/2) in the Pythagorean triangle where θ is the angle opposite the side of length a. The reason that sometimes we have to divide by two to make the Euclid-generated Pythagorean triple be primitive is that the focus on tan(θ/2) is in some sense inappropriate. If we instead look at tan(θ) = a/b in lowest terms (or look at sin(θ) = a/c or cos(θ) = b/c in lowest terms) we would have two-thirds of our primitive Pythagorean triple immediately, never having to divide by 2. Fortunately, the double-angle formula for a tangent is easy enough to analyze; tan(θ) = 2 tan(θ/2) / (1 - tan2(θ/2)) = 2mn / (m2 - n2) will be in lowest terms exactly when n and m are relatively prime and not both odd. If they are relatively prime but both odd then the lowest terms representation for tan(θ) requires that both the numerator 2mn and denominator m2 - n2 be halved. 𝕃eegrc ( talk) 16:29, 6 October 2016 (UTC)
When m and n are coprime but both odd then (m′, n′) = ((m+n)/2, (m−n)/2) will be coprime, not both odd, and will generate the primitive Pythagorean triple associated with (m, n), but with a and b reversed. This follows, in part, from the fact that the two non-right angles in a Pythagorean triangle sum to π/2, so their half angles sum to π/4, and the formula tan(π/4 - θ/2) = (1 − tan(θ/2)) / (1 + tan(θ/2)). 𝕃eegrc ( talk) 15:53, 7 October 2016 (UTC)
There are various parts of the article that assume as a pre-established fact that a is odd and b is even or vice versa. This fact is never established This fact is assumed before it has been established, and is not clearly enough described to justify how much it's referred to. The fact that either a or b is even needs to be made more prominent.
Pythagorean triple/Archive 5#Generating a triple - image to the right:
The primitive Pythagorean triples. The odd leg a is plotted on the horizontal axis, the even leg b on the vertical."
Pythagorean triple/Archive 5#General properties:
The properties of a primitive Pythagorean triple (a, b, c) with a < b < c (without specifying which of a or b is even and which is odd) include:
Pythagorean triple/Archive 5#Proof of Euclid's formula: "As a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd (if b were odd, c would be even, and c2 would be a multiple of 4, while a2 + b2 would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4)."
-- NeatNit ( talk) 18:00, 4 May 2017 (UTC)
It would be nice if a chart like this colored triples differently whether they were primitive or not. Tom Ruen ( talk) 02:57, 31 August 2017 (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 147.215.1.189 ( talk) 12:43, 6 October 2015 (UTC)
In the article that is not choosen as a definition, but instead gcd(a,b,c) = 1 : coprime as a set. This not the same! For a ppt one can prove that the above pairwise coprime properties hold. This is done later in the article. I've tried to clear this up. Albert 80.100.243.19 ( talk) 20:29, 1 November 2017 (UTC)
In the § Special cases section, these three consecutive bullet points need either a simple explanation or references, as appropriate:
- There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 202 + 212 = 292; these are generated by Euclid's formula when is a convergent to √2.
- For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
- For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number. Done
yoyo ( talk) 13:23, 24 September 2018 (UTC)
Pinging @ Mifter and Widr: who helped deal with the last disruption: the same person is back with a different IP address. This IP has a more prolific recent history, some of which is similar vandalism. I am not able to check whether the edits to articles about skyscrapers in Taiwan are legitimate. -- JBL ( talk) 15:43, 2 January 2019 (UTC)
Hi! See "the largest number in a Pythagorean triple". For F9 = 34 this Pythagorean triple is (30, 16, 34) = 2 * (15, 8, 17). So "the largest number in a Pythagorean triple" could be expanded to "the largest number in a Pythagorean triple which must not be a primitive Pythagorean triple".
Regards
Gangleri
11:56, 9 July 2019 (UTC) — Preceding
unsigned comment added by
217.86.249.144 (
talk)
It may be true that what follows has been published, but, if so, I haven't seen it. (Or, because I'm either (a) too lazy or (b) too innumerate to decipher all of the article about Pythagorean Triples, it may be described in that article.) Using an Excel spreadsheet and some tinkering with known primitive triples, I came up with the following four series or formulas for developing Pythagorean Triples.
Formulas for creating Pythagorean Triples
If there is another formula for deriving Pythagorean Triples other than the four above, I do not know it. The most common way to derive a P.T. as stated in most math textbooks is to choose two different counting numbers m and n, then apply the following expressions to obtain the three side lengths: |m^2-n^2| , 2mn, and m^2 + n^2. My gut feeling, not proven, is that no matter what two numbers are chosen, the resulting P.T. will be found in one of the four series described above. Also, it appears that, if the difference between m and n is odd, the triple formed will be primitive.
72.243.144.96 ( talk) 02:57, 10 July 2019 (UTC) Tom Hobbs
This
edit request to
Pythagorean triple has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
Please vikify the first ocurence of hypotenuse. Thanks in advance no bias — קיין ומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contributions 18:51, 17 July 2019 (UTC)
Hi! I think the article should emphasize the unique properties of the primitive Pythagorean triple(t)s. What I try to tell: "Exactly one of a, b is divisible by 3." (3, 4, 7) has this property too but it is not a Pythagorean triplet. regards no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 02:12, 19 July 2019 (UTC)
Titus III has recently made an addition ( once, twice) concerning the "upside-down Pythagorean theorem". The obvious problem with it is that it refers to the equation as a theorem, and this is obviously wrong: it's missing all the associated data (what do those symbols x, y, z represent?) that one needs to make sense of it. I think inclusion in some form is possible; in accordance with WP:BRD, I propose we discuss here to refine it first. -- JBL ( talk) 18:41, 16 August 2019 (UTC)
"The upside down Pythagorean theorem, [1] [2]
gives the relationship of the two legs of a right triangle to the altitude (a line from the right angle and perpendicular to the hypotenuse). [3] The equation can be transformed to,
and is the Pythagorean triple . If the are to be integers, the smallest solution is then
using the smallest Pythagorean triple ." Titus III ( talk) 19:14, 16 August 2019 (UTC)
References
Quoting the section on properties: "For each natural number n, there exist at least n different Pythagorean triples with the same hypotenuse.[8]:31"
It is not clear what is meant by this. I interpreted it as "with n as hypotenuse", which is clearly wrong (There is only one triplet with 5 as hypotenuse: (3,4,5)). Otherwise, what is that "same hypotenuse"? Is this just a difficult way of stating that there exist inifinitely many Pythagorean triples? 37.44.138.159 ( talk) 10:53, 9 September 2019 (UTC)
I read the pragraphe of "historic . It s a shame that in an scientfic article , you mix some semitic revendications nationalist and fascist If i read the article about "plimpton 322" there are some tones of theories and nobody agrees how to translate this babylonian document . Some theories tell t that the document was a table of reciprocal numbers , some other tell it was trigonometry , some other , it was a commercial transation document . Is it serious ? Which one is the good ?
So why should it be marked i as historic in this page ? Tio make please to fascists ? In addition , why this historic doesn t make mention about all the other dicoveries in the history about pythagorician triplets ? Is it shameful for semitics ? Miranda2016 ( talk) 15:46, 16 January 2020 (UTC)
k in the lede for "if (abc) is a triple, then so is (ka kb kc) but s in the diagram. -- 142.163.195.153 ( talk) 16:55, 25 February 2021 (UTC)
Thank you @ DVdm: for looking out for encyclopedic concerns. All, if you have a citation that backs the following, please supply it here.
Thank you — Quantling ( talk | contribs) 18:57, 23 March 2021 (UTC)
From WP:LEAD "Apart from basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article." Right now, the lead contains an extended definition that does not exist in the article body. I just tried to remedy this but was reverted. I invite suggestions and how to fix this. LK ( talk) 06:36, 9 April 2021 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 3 | Archive 4 | Archive 5 |
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)
The following was put into the article's section on Platonic sequences on 28 June 2007 by someone who is no longer on Wikipedia:
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:
Loraof ( talk) 17:58, 8 April 2015 (UTC)
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 128.84.127.40 changed it. — Anita5192 ( talk) 21:14, 29 January 2016 (UTC)
Some triplet comes also from:
A=3, B=4, C=5 is the most famous. — Preceding unsigned comment added by 94.81.217.96 ( talk) 13:15, 19 May 2016 (UTC)
It seems reasonable to add a link to /info/en/?search=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)
Done D.Lazard ( talk) 07:52, 6 October 2016 (UTC)
In the text it says that "A proof of the necessity that be expressed by Euclid's formula for any primitive Pythagorean triple is as follows". However, Euclid's formula only gives primitive Pythagorean triples with even. It is for instance obvious that we cannot find m and n such that , , . As pointed out earlier, if is even and , Euclid's formula gives where is a PPT. For instance and gives .
So a few places, primitive Pythagorean triple should be replaced by primitive Pythagorean triple where is even. — Preceding unsigned comment added by HelmerAslaksen ( talk • contribs) 19:44, 6 September 2016 (UTC)
HelmerAslaksen ( talk) 00:34, 6 October 2016 (UTC)
Take a look at Pythagorean triple#Interpretation of parameters in Euclid's formula. There we see that n/m is the lowest terms representation of tan(θ/2) in the Pythagorean triangle where θ is the angle opposite the side of length a. The reason that sometimes we have to divide by two to make the Euclid-generated Pythagorean triple be primitive is that the focus on tan(θ/2) is in some sense inappropriate. If we instead look at tan(θ) = a/b in lowest terms (or look at sin(θ) = a/c or cos(θ) = b/c in lowest terms) we would have two-thirds of our primitive Pythagorean triple immediately, never having to divide by 2. Fortunately, the double-angle formula for a tangent is easy enough to analyze; tan(θ) = 2 tan(θ/2) / (1 - tan2(θ/2)) = 2mn / (m2 - n2) will be in lowest terms exactly when n and m are relatively prime and not both odd. If they are relatively prime but both odd then the lowest terms representation for tan(θ) requires that both the numerator 2mn and denominator m2 - n2 be halved. 𝕃eegrc ( talk) 16:29, 6 October 2016 (UTC)
When m and n are coprime but both odd then (m′, n′) = ((m+n)/2, (m−n)/2) will be coprime, not both odd, and will generate the primitive Pythagorean triple associated with (m, n), but with a and b reversed. This follows, in part, from the fact that the two non-right angles in a Pythagorean triangle sum to π/2, so their half angles sum to π/4, and the formula tan(π/4 - θ/2) = (1 − tan(θ/2)) / (1 + tan(θ/2)). 𝕃eegrc ( talk) 15:53, 7 October 2016 (UTC)
There are various parts of the article that assume as a pre-established fact that a is odd and b is even or vice versa. This fact is never established This fact is assumed before it has been established, and is not clearly enough described to justify how much it's referred to. The fact that either a or b is even needs to be made more prominent.
Pythagorean triple/Archive 5#Generating a triple - image to the right:
The primitive Pythagorean triples. The odd leg a is plotted on the horizontal axis, the even leg b on the vertical."
Pythagorean triple/Archive 5#General properties:
The properties of a primitive Pythagorean triple (a, b, c) with a < b < c (without specifying which of a or b is even and which is odd) include:
Pythagorean triple/Archive 5#Proof of Euclid's formula: "As a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd (if b were odd, c would be even, and c2 would be a multiple of 4, while a2 + b2 would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4)."
-- NeatNit ( talk) 18:00, 4 May 2017 (UTC)
It would be nice if a chart like this colored triples differently whether they were primitive or not. Tom Ruen ( talk) 02:57, 31 August 2017 (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 147.215.1.189 ( talk) 12:43, 6 October 2015 (UTC)
In the article that is not choosen as a definition, but instead gcd(a,b,c) = 1 : coprime as a set. This not the same! For a ppt one can prove that the above pairwise coprime properties hold. This is done later in the article. I've tried to clear this up. Albert 80.100.243.19 ( talk) 20:29, 1 November 2017 (UTC)
In the § Special cases section, these three consecutive bullet points need either a simple explanation or references, as appropriate:
- There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 202 + 212 = 292; these are generated by Euclid's formula when is a convergent to √2.
- For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
- For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number. Done
yoyo ( talk) 13:23, 24 September 2018 (UTC)
Pinging @ Mifter and Widr: who helped deal with the last disruption: the same person is back with a different IP address. This IP has a more prolific recent history, some of which is similar vandalism. I am not able to check whether the edits to articles about skyscrapers in Taiwan are legitimate. -- JBL ( talk) 15:43, 2 January 2019 (UTC)
Hi! See "the largest number in a Pythagorean triple". For F9 = 34 this Pythagorean triple is (30, 16, 34) = 2 * (15, 8, 17). So "the largest number in a Pythagorean triple" could be expanded to "the largest number in a Pythagorean triple which must not be a primitive Pythagorean triple".
Regards
Gangleri
11:56, 9 July 2019 (UTC) — Preceding
unsigned comment added by
217.86.249.144 (
talk)
It may be true that what follows has been published, but, if so, I haven't seen it. (Or, because I'm either (a) too lazy or (b) too innumerate to decipher all of the article about Pythagorean Triples, it may be described in that article.) Using an Excel spreadsheet and some tinkering with known primitive triples, I came up with the following four series or formulas for developing Pythagorean Triples.
Formulas for creating Pythagorean Triples
If there is another formula for deriving Pythagorean Triples other than the four above, I do not know it. The most common way to derive a P.T. as stated in most math textbooks is to choose two different counting numbers m and n, then apply the following expressions to obtain the three side lengths: |m^2-n^2| , 2mn, and m^2 + n^2. My gut feeling, not proven, is that no matter what two numbers are chosen, the resulting P.T. will be found in one of the four series described above. Also, it appears that, if the difference between m and n is odd, the triple formed will be primitive.
72.243.144.96 ( talk) 02:57, 10 July 2019 (UTC) Tom Hobbs
This
edit request to
Pythagorean triple has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
Please vikify the first ocurence of hypotenuse. Thanks in advance no bias — קיין ומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contributions 18:51, 17 July 2019 (UTC)
Hi! I think the article should emphasize the unique properties of the primitive Pythagorean triple(t)s. What I try to tell: "Exactly one of a, b is divisible by 3." (3, 4, 7) has this property too but it is not a Pythagorean triplet. regards no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 02:12, 19 July 2019 (UTC)
Titus III has recently made an addition ( once, twice) concerning the "upside-down Pythagorean theorem". The obvious problem with it is that it refers to the equation as a theorem, and this is obviously wrong: it's missing all the associated data (what do those symbols x, y, z represent?) that one needs to make sense of it. I think inclusion in some form is possible; in accordance with WP:BRD, I propose we discuss here to refine it first. -- JBL ( talk) 18:41, 16 August 2019 (UTC)
"The upside down Pythagorean theorem, [1] [2]
gives the relationship of the two legs of a right triangle to the altitude (a line from the right angle and perpendicular to the hypotenuse). [3] The equation can be transformed to,
and is the Pythagorean triple . If the are to be integers, the smallest solution is then
using the smallest Pythagorean triple ." Titus III ( talk) 19:14, 16 August 2019 (UTC)
References
Quoting the section on properties: "For each natural number n, there exist at least n different Pythagorean triples with the same hypotenuse.[8]:31"
It is not clear what is meant by this. I interpreted it as "with n as hypotenuse", which is clearly wrong (There is only one triplet with 5 as hypotenuse: (3,4,5)). Otherwise, what is that "same hypotenuse"? Is this just a difficult way of stating that there exist inifinitely many Pythagorean triples? 37.44.138.159 ( talk) 10:53, 9 September 2019 (UTC)
I read the pragraphe of "historic . It s a shame that in an scientfic article , you mix some semitic revendications nationalist and fascist If i read the article about "plimpton 322" there are some tones of theories and nobody agrees how to translate this babylonian document . Some theories tell t that the document was a table of reciprocal numbers , some other tell it was trigonometry , some other , it was a commercial transation document . Is it serious ? Which one is the good ?
So why should it be marked i as historic in this page ? Tio make please to fascists ? In addition , why this historic doesn t make mention about all the other dicoveries in the history about pythagorician triplets ? Is it shameful for semitics ? Miranda2016 ( talk) 15:46, 16 January 2020 (UTC)
k in the lede for "if (abc) is a triple, then so is (ka kb kc) but s in the diagram. -- 142.163.195.153 ( talk) 16:55, 25 February 2021 (UTC)
Thank you @ DVdm: for looking out for encyclopedic concerns. All, if you have a citation that backs the following, please supply it here.
Thank you — Quantling ( talk | contribs) 18:57, 23 March 2021 (UTC)
From WP:LEAD "Apart from basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article." Right now, the lead contains an extended definition that does not exist in the article body. I just tried to remedy this but was reverted. I invite suggestions and how to fix this. LK ( talk) 06:36, 9 April 2021 (UTC)