Talk:Euclid's lemma

message: "It states that if a prime divides the product of two numbers, it must divide at least one of the factors" <-- It's ambiguous whether or not the prime must divide one of the two numbers we multiplied, or if it must divide any one of the factors of the resulting number (which are potentially much more than the two we originally multiplied).  — Preceding unsigned comment added by 38.98.147.133 (talk) 20:04, 11 October 2012 (UTC)
thread_title:  Issue with intro wording 

message: With the exception of one link to google books, there are no references in the paper at the moment. It might be good to add some. (In particular, I was not able to locate the alternative proof in any textbooks.) --Kompik (talk) 09:46, 24 July 2011 (UTC)
thread_title: Unreferenced template

message: I have rewritten the article. It now has references. I replaced the "Alternative proof" with a "Euclidean proof". - Virginia-American (talk) 17:13, 5 July 2012 (UTC)
thread_title:  rewrite 

message: Euclid's lemma is based on Euclid's algorithm. Euclid's algorithm has to be embedded in theorem 1 of Euclidean number theory. The properties and the name encluded in the embedding will be clear if you apply the algorithm. I wrote a good example in the talk page of the site Euclid's algorithm. If you like I can edit another one in your article. Most sites don't use this algorithm, which is a pity and a source of endless twaddle.
Theorem 1 : [Euclid's algorithm] Each pair a, b ε Z+ have a gcd (a, b). This gcd (a, b) can be calculated by Euclid's algorithm and has the following propertiies. (i) Gcd (a, b) ε Ζ+. (ii) Gcd (a, b) is a common divisor of a and b. (iii) Each common divisor of a and b is a divisor of gcd (a, b). (iv) The equation ax + by = gcd (a, b) has a solution in Z. The properties (ii) and (iii) explain the name greatest common divisor.
Theorem 2 : [Euclid's lemma] p prime and p | ab -----> p | a or p | b.
Given p prime, p | ab and assume p is not a divisor of a. Then gcd (p, a) = 1, because the only divisors of p are 1 and p himself but p is by assumption not a divisor of a. Now applying Euclid's algorithm [point (iv)] one concludes : the equation px + ay = gcd (p, a) = 1 has a solution for x and y in Z+. Multiplying this equation by b one gets bpx + bay = b. In this equation p | bpx and p | (ab)x [given]. Thus p | b [their sum]. End.C.W. Vugs (talk) 15:01, 13 April 2013 (UTC)
thread_title:  OLD PROOF 

message: I find the entire "Euclidean proof" section to be ridiculously excessive. The sheer size of the section gives entirely the wrong impression and is likely to put off casual readers. The only bits that are actually relevant are the last three lines. Just because we want to include Euclid's original proof doesn't mean we have to include every single lemma and proposition leading up to it, by that logic the article on the Pythagorean theorem would include most of Book I of the Elements.
It also buries the artistry in the actual proof. What intermediate steps are embedded in the proof of a theorem, versus which ones are separated out into lemmas, is an expression of the mathematician's feelings on how the mathematical structures under study fit together, and what is the most elegant way of understanding them.
The entire section could be shortened to this:
Since a|bc, a ≠ 0. If b = 0, gcd(a, 0) = 1 implies that a = ±1 . But then a|c. If b ≠ 0, let m = lcm(a, b). Since gcd(a, b) = 1, m = |ab|. Since a|bc and b|bc, bc is a common multiple of a and b. Therefore m|bc, ab|bc, a|c.
The only non-trivial fact that this relies on is the relationship between LCM and GCD, which is elementary number theory. I'm going to make the change, please discuss it here if you're opposed. Gaiacarra (talk) 21:22, 19 September 2013 (UTC)
thread_title:  "Euclidean proof" is overkill 

message: In  this edit reference for the alternative name "Euclid's first theorem" was requested I have added  one book] where I found this results under this name. If you know other (perhaps more appropriate) references, please, edit the article. --Kompik (talk) 06:24, 25 April 2015 (UTC)
replies: text: It's a bad name (because it is very far from being the first name in Euclid) but it does indeed appear to be called that. I added another reference, as well as a note pointing out the badness of the name and giving a reference to the result in geometry that is actually Euclid's first theorem. —David Eppstein (talk) 06:45, 25 April 2015 (UTC)
thread_title: Also called Euclid's first theorem

message: This article needs help from someone who is both a subject matter expert in the field and understands English syntax. Some sentences are too messed up for a non-SME to decode into clear prose.2602:306:CDD1:2D00:7076:A115:27CB:15C9 (talk) 20:23, 18 December 2016 (UTC)
replies: text: It's not subject-matter expertise that is the issue, it's the fact that we're directly quoting from a source that was written in a somewhat old-fashioned style of English. Please do not edit direct quotes, as you did, in an attempt to clean up their grammar. —David Eppstein (talk) 21:03, 18 December 2016 (UTC)
thread_title:  Garbled sentences 

message: Proposition 19 states:
If four numbers be proportional, the number produced from the first and fourth is equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers are proportional.
There is a note at the end of this which states: 
If a ： b＝c ： d, then ad＝bc; and conversely
This is blatantly false, consider the simplest example consisting of x,2x,3x, and 4x. All of them are proportional, but the product of the first and forth is 4x^2, while the second and third produces 6x^2.
I think something has been misquoted here. — Preceding unsigned comment added by 49.199.23.115 (talk • contribs) 09:22, 18 September 2020 (UTC)
replies: text: It is not true that x:2x = 3x:4x. The first of these two ratios is 1:2 and the second is 3:4, not equal. So this is not a counterexample. —David Eppstein (talk) 16:28, 18 September 2020 (UTC)
thread_title:  Proposition 19 is false in its current wording  

message: hi, i just wanted to know if the euclid's division lemma https://www.toppr.com/content/concept/euclids-division-lemma-206838/ is same as euclid's lemma. The page named 'euclid's division lemma' does not exist in wikipedia. Is it known by some other name, or it has to be created. Huzaifa abedeen (talk) 03:54, 30 September 2020 (UTC)
replies: text: Euclid's division lemma exists in Wikipedia, and is different from Euclid's lemma. D.Lazard (talk) 08:07, 30 September 2020 (UTC)
thread_title:  Euclid's division lemma