C.W. Vugs

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Good luck and happy editing. ```Buster Seven Talk 13:01, 16 November 2012 (UTC)

a² + b² = m² + n² = c²

Do you happen to know any formula(s) that can generate ALL bi-pythagorean (or pluri-pythagorean) numbers ? By that I mean numbers whose square can be written as the sum of two other squares in two (or more) different ways ? — Preceding unsigned comment added by 79.118.170.244 (talk) 14:08, 13 February 2013 (UTC)

Nevermind, I think I found the answer. Apparently, this only seems to happen if the root itself contains among its factors or divisors two (or more) distinct Pythagorean numbers of the form 4k + 1:

D_C = { C₁, C₂, ... C_m } <=> C² = A₁² + B₁² = A₂² + B₂² = ... = A_n² + B_n² , where m ≤ n, and A_i ≠ A_j , B_i ≠ B_j for all and any i ≠ j.

— Preceding unsigned comment added by 79.113.231.13 (talk) 19:06, 15 February 2013 (UTC)

Talk:Pythagorean_triple

your comment drew my attention. I don't have a general formula but I just posted into www.mymathforum.com observations of the prime hypotenuse H. Essentially, H(n)/Prime(n) ~ 2.15 (the first 10000 tend towards this limit). For example, take Prime(500000)=7368787. We expect a PPT near 2.15*7368787 ~ 15842892. We find the PPT at 15843629 (ratio 2.1501...).--Billymac00 (talk) 00:23, 18 March 2013 (UTC)

Talk pages and Pythagorean triple

Hi C.W. Vugs,

I wanted to draw your attention to the page Help:Introduction to talk pages. Article talk pages are for discussing changes to articles, not for general discussion of the topic of the article or as a place to publish personal research. You have been adding a large amount of content to Talk:Pythagorean triple, and it's not obviously related to any proposed edits to the article.

Happy editing, --JBL (talk) 19:16, 25 March 2013 (UTC)

CRITICAL REMARKS

Good day Joel I appreciate that I can talk to a person and not to a number. Here a list of critical remarks related to the article Pythagorean triple.

(i) The definition of a PPT : the claim a, b, c are pairwise coprime (or the same gcd(a,b) = gcd(b,c) = gcd(c,a) = 1) is abundant.Gcd(a,b) = 1 or, more commonly used, gcd(a,b,c) = 1 is sufficient. The following fact can easily be proved : (a,b,c) a PPT —> gcd(a,b) = gcd(b,c) = gcd(c,a) = 1. A good definition : (a,b,c) a PPT <—> a, b, c ε Z⁺, a² + b² = c² and gcd(a,b,c) = 1.

(ii) Section : generating a triple. The text If both m and n are odd, then a, b, c will be even, and so the triple will not be primitive. Correct, because the same is true when both m and n are even. The continuation of the sentence however, dividing a, b, c by 2 will yield a primitive triple if m and n are copprime. Not correct; here you use that gcd(m,n) = 1 while you have to prove that. It is given (a,b,c) is a PPT and gcd(m,n) = 1 has to be proved, moreover one can't divide a, b, c by 2. Two separated facts has to be proved : (a) (a,b,c) is a PPT —> m and n are impair (this is done) and (b) (a,b,c) is a PPT —> gcd(m,n) = 1. This is all but trivial. You can't avoid proofs by contradiction.

(iii) Section : proof of Euclid's formula. The easy parts m, n ε Z⁺, m > n ?, and (m² - n²)² + (2mn)² = (m² + n²)² are proved but not the difficult fact gcd(m,n) = 1 —> gcd(m² - n², 2mn, m² + n²) = 1. This is the inverse of (iib) and also not trivial.

The necessity part is correct but not simple at all, as announced in the beginning. Especially the last part can be simplified. In the meantime the reader will be lost in the a-b-c-m-n magic and the story should be resumed in a theorem : (a,b,c) a PPT <—> there are m and n ε Z⁺, m > n, m and n impair, gcd(m,n) = 1 and (m² - n², 2mn, m² + n²) = (a, b, c).

(iv) Section : interpretation of parameters in Euclid's can better be moved to the section some relationships. Give here two simple examples and work them out to show how the m,n-substitution is used and how the created diophantine equation is solved. For example (i) find two PPT's with a leg of 36 en (ii) find one PPT with a perimeter of 208.

(v) Section : elementary properties of PPT's. This is a good section and the reason that this article is frequently visited. Please change the heading and keep b even. In the proof of Euclid's formula b is even, in 20² + 21² = 29² b is odd, in a + j² = c = b + 2^k b is odd and in the last but one property b is even.

Point 3 reads : at most one of a, b, c is a square. This is a hard one. The article refers to [[Infinite descent#Non-solvability of R⁴ + S⁴ = T⁴]]. The proof in that site is ridiculous. Better is [Proof of Fermat's Last Theorem for specific exponents] or my contribution in the talk page.

Point 19 reads : all prime factors of c are primes of the form 4n + 1. This one seems to be original. It is for sure a very hard one. I am struggling with this one and like to have references or other help to prove it.

(vi) Section : some relationships. At the end one reads : If two numbers of a triple are known, the third can be found using the Pythagorean theorem. This is the Pythagorean theorem of grade 7. The following has to be added : If one number of a triple is known , the two others can be found using Euclid's formula. This is very important and shows the sense and utility of Euclid's formula. A diophantine equation of three unknowns can be reduced to two unknowns by the m,n-substitution. The article makes no use of this.

(vii) Section : the platonic sequence. In the blackboard part side a is mixed with some even or odd number a and I think nobody understand what is going on. See for this my contribution on the talk page.

P.S. So far I went through the article. Pythagorean triple is a very interesting subject. Greetings. C.W.VugsC.W. Vugs (talk) 16:00, 26 March 2013 (UTC)

To be honest, I think that the article has to be rewritten up to and including platonic sequence. The properties can stay with some little changes.C.W. Vugs (talk) 09:56, 28 March 2013 (UTC)

Do not use talk pages for blogging

I have removed your last edit in talk: Euclidean algorithm because this page is devoted to discussion on how to improve the article. It must not be used for blogging or exposing your own views on the subject (see WP:talk and WP:NOTAFORUM)

My view is that you not give enough attention to Euclid's algorithm and the GCD conception, defined inside the theorem of this algorithm. This is the basic of Euclidean number theory. This I like to address by giving examples rather then slogans about fields, skew fields, rings, Euclidean domains, ideals, principal ideals and what more. Number theory up to and included Fermat (ca 1650) was full of GCD statements and proofs by contradiction and I see nothing of this in Wikipedia's articles about Euclidean number theory. This is impossible and can not be replaced by modular arithmetic. C.W. Vugs (talk) 19:16, 10 May 2013 (UTC)