Talk:Bézout's identity

message: I fixed the proof for two co-primes.  I am new to this and may have made many non-standard math formatting, etc.  If you can review, please do.
Thanks (SLB December 2016)  — Preceding unsigned comment added by Sandybultena (talk • contribs) 02:28, 14 December 2016 (UTC)
thread_title: New Proof

message: I should do the searching... but just in case. Is it true that one can get the coefficients of less degree than the starting elements if one is in an Euclidean Domain? In the case of integers this amounts to saying that
there are x,y ∈ Z wiht |x|<max(a,b), |y|<max(a,b) with xa+yb=gcd(a,b)
Just in case. I'll try to find out though :) Pfortuny 09:26, 20 Apr 2004 (UTC)
thread_title: Untitled

message: By the way, does anyone know why this is called an "identity"?  Usually an identity consists of two expressions which are always equal; but, here, what we've got is a statement that a certain (simple) Diophantine equation has a solution.  JadeNB 01:59, 29 April 2007 (UTC)
replies: text: I believe the identity is that the set of linear combinations of x & y is identical to the set of multiples of gcd(x,y). Blokhead 01:32, 14 May 2007 (UTC)
thread_title:  Name 

message: Well this one seemed more rigorous to me:
http://www.cut-the-knot.org/blue/Euclid.shtml
Essentially works backwards from Euclids algo to get here.
Does anyone know a way to prove this using co-ordinate geometry?
Note that this is the equation of a line
given:ax+by=gcd(a,b),  show that integers x,y exist on this line.
However note that this too shows that a solution of the form
ax+by=gcd(a,b) exists
It does not however show that gcd(a,b) is the smallest possible solution.  —Preceding unsigned comment added by 220.227.207.194 (talk) 07:01, 2 April 2008 (UTC)
thread_title:  nice proof 

message: According to my paper no. 2 , VOL. No. 2, published on my journal ( http://nilaish.livejournal.com/ ), It is now stated that my proof is much simpler. I'll try to explain it here, we have,
Linear Diophantine Equation ,
ax + by = c , is general Linear Diophantine Equation, as per my solution its general solutions are,
x = (c^2 + 1)/2ac ; y = (c^2 - 1)/2bc , as we notice when c = 1 , then LDE has infinite integral solutions, c other than 1, it has non integral solutions. So, my this citation is much simpler than the provided proof. To know more about my paper no. 2, please visit my journal. I think that this can be  treated as considerable substance to treat various LDEs, which can be verified using my formulae. —Preceding unsigned comment added by Nilaish (talk • contribs)  15:32, June 20, 2008 
:Hmmm ... something clearly wrong there, because when c = 1 you have y = 0 and x = 1/a, which is only an integral solution in the special case when a is 1. Gandalf61 (talk) 10:32, 21 June 2008 (UTC)
thread_title:  A simple Proof of Bezout Identity by Nilaish 

message: I think I understand the identity and the solutions but I don't understand this section:
 
  ... then the K'th ratio in this set will be:
  (k19 + 5)/(k15 + 4)   k=0, 1, 2 ...
Can someone please fill in some detail on this and how the "approximation" part comes in? 
What "set" does this refer to? The multivalued solution set for x, y?
Jensigner (talk) 02:15, 23 August 2010 (UTC)
replies: text: I think I don't understand the musical intervals section at all, nor apparently does the person who wrote it. Can anyone who understands what this is supposed to be about rewrite it in understandable language? Maybe it would be an idea to start explaining what the problem is with musical intervals in the first place. If this cannot be done, I think the section should be removed. Marc van Leeuwen (talk) 21:18, 3 March 2011 (UTC)
replies: text: I can't understand the point of that section either. It adds nothing useful to the article, so I have removed it. Gandalf61 (talk) 10:07, 4 March 2011 (UTC)
thread_title:  Musical Intervals example 

message: To anybody with a background in musical theory, it is "goes without saying" that the GCD between two or more sine waves corresponds to the frequency of that wave. Furthermore, if we stay within the realm of small whole numbers (the tuning system called Just Intonation or JI), then this GCD acts as the Tonic or key centre in musical harmony. The Bezout's ID technique I described has been used as a standard method for finding approximate musical intervals for centuries. 
The fact that a few 'pure' mathematicians believe they can decide for the rest of the world what is of universal relevance is arrogant to say the least. Since 1928 it has been known that the universe is wave-like and is not based on infinitesimal points of zero dimensions. Indeed, questions like "What "set" does this refer to?", as if wave theory could ever be reduced to a problem of non-existent number sets, could be taken as a measure of the extent to which 'pure' maths has allowed itself to be removed from reality. But at the end of the day, it was really none of your business to decide amongst yourselves to delete my article, based as it was on your own lack of understanding. I'll get back to it when I'm ready.  —Preceding unsigned comment added by 122.108.152.136 (talk) 08:32, 28 March 2011 (UTC)
replies: text: You say "it was really none of your business to decide amongst yourselves to delete my article" - but, actually, it is entirely the business of other editors to decide whether to keep, amend or remove another editor's contributions, because Wikipedia is a collaborative venture. There is no such thing as "my article" - see WP:OWN. The musical intervals example was removed because no-one could understand it - see discussion above. If you are going to reinstate it, you should try to explain it more clearly, and provide a reliable source to show that it is not your own original research. Gandalf61 (talk) 15:31, 28 March 2011 (UTC)

text: As someone who is both a 'pure' mathematician and has some background in musical theory, let me try to say something about the musical contents of the remark. When you say "GCD between two or more sine waves" this is at first sight unclear: GCD operates on integers (or on polynomials, but I don't think that is relevant her), sine waves are functions of a real variable. What you probably mean is taking the GCD of their frequencies (which makes the remainder of the sentence "corresponds to the frequency of that wave" unclear, but it is so anyway since there were multiple waves). So I think you are saying that the GCD of two or more frequencies gives a "ground" frequency of which all of them are harmonics; this is true, more or less by definition of GCD and of harmonics. When you say this GCD acts as the Tonic or key centre in musical harmony, I don't agree fully. It is true in the case of three frequencies related respectively 4:5 and 5:6, i.e., a major chord: these are three harmonics of a note two octaves below the first frequency, which indeed is the key centre for the major chord. It is not true however for three frequencies related respectively 5:4 and 6:5, i.e., a minor chord; if one seeks three integers having these successive ratios then the smallest instance is 10,12,15, which means the GCD is the frequency for which these are harmonics number 10, 12, and 15; concretely for the chord A4,C5,E5 it would be F2 a note that is not the key centre for the minor chord. Which is just to say, you should be careful to state precisely what you mean, otherwise people will not be able to understand. I'm sure there is some sense to the "Bezout's ID technique" you described, but from the given description I could not understand either what precisely is the problem to be solved, what the technique consist of, and why it solves the problem. If it is as old as you claim, then certainly it can be already found somewhere on the wikipedia, more likely in the mucic-oriented pages than in the mathematical pages. A clear description with reference to the related musical topics would be welcome. But something that is only clear to the author of the text has very little use here. Marc van Leeuwen (talk) 07:23, 29 March 2011 (UTC)
thread_title: Why was my section on musical intervals deleted?

message: The following is unclear.  
:In addition d > 0 is their greatest common divisor and the smallest positive integer that can be written, in this form, for any integers x,y.
The first sentense says that it always holds *if* d is some common divisor, and then the above sentense says it must be the GCD, if it says anything at all.  Which is it,GCD or a divisor? 
(I have tidied up the other confusing language but left this.  Maths articles should be intelligible to people that do not already understand the article's content before reading it!)
Tuntable (talk) 03:40, 4 June 2012 (UTC)
thread_title:  Introduction does not make sense 

message: Hi all,
I was looking for the identity and I came to this page. I believe the first statement (sentences 1 and 2) is actually false. If d is a divisor of a and b, Bezout's coefficients exist iff d is also the gcd of a and b. In particular, it is not sufficient for d to be a divisor of both a and b. As an example, you may take a = b = 4 and d = 2.
Should we change it?  — Preceding unsigned comment added by Pinoeottavio (talk • contribs) 09:17, 22 June 2012 (UTC)
replies: text: If you read the first paragraph carefully, you will see that it doesn't claim that Bezout's identity is satisfied for any common divisor of a and b. In the second paragraph it goes on to say that if Bezout's identity is satisfied for some common divisor d > 0 then d is the gcd of a and b. Gandalf61 (talk) 10:40, 22 June 2012 (UTC)
thread_title:  False statement? 

message: I really struggled with the proof, and judging from the other comments I think I'm not alone. I have trouble making heads or tails from the one on the page. I thikn it's important enough to make much more accessible. Combined with the Gandalf61's intuitive explanation, I found the one here to be excellent. Would anyone mind if I began beefing up the proof section using these resources? Garfieldnate (talk) 05:20, 8 April 2013 (UTC)
thread_title:  Proof Fixup 

message: In the introduction (before the contents list), it is stated:
:A pair of Bézout coefficients (in fact the ones that are minimal in absolute value) can be computed by the extended Euclidean algorithm.
but no source for the minimality of the Bézout coefficients (let alone the intended meaning of minimality in absolute value for pairs of numbers) is given, nor could I find one in the article itself or the linked article on the extended Euclidean algorithm.  Lambdacalculator (talk)  —Preceding undated comment added 19:34, 1 November 2013 (UTC)
replies: text: In fact, the extended Euclidean algorithm does not product always the minimal pair, but one of the two minimal pairs. I have corrected the assertion. For a reference, I do not know, but the reference book for this kind of results is "Knuth, The art of computer programming, Seminumerical algorithms". The proof is as follows, supposing that a and b are positive: Let q_i and r_i be the successive quotients and remainders in Euclidean algorithm. x and y are obtained recursively by x_0=1, x_1=0, x_{i+1}=x_{i-1}-q_i x_i and y_0=0, y_1=1, y_{i+1}=y_{i-1}-q_i y_i. As r_{i-1}=r_{i+1}+q_i r_i and  r_{i+1}>0, we have q_i<\frac{r_{i-1}}{r_i}. It follows that the x_i and the y_i are increasing in absolute value, change of sign at each step and are such that \left |\frac {x_{i}}{x_{i-1}}\right |< q_i and \left |\frac {y_{i}}{y_{i-1}}\right |< q_i. The result is obtained by taking the product of these inequalities when i varies. D.Lazard (talk) 21:48, 1 November 2013 (UTC)
thread_title:  Source for minimality claim in the intro? 

message: The article writes  |x| < |b|/|d|  ...
But for  a = 2,  b = 0,  this inequality occurs   |x| < 0,  -- a contradiction.
May be, it is needed to require  "both non-zero" ?
Mechvel (talk) 13:35, 23 January 2014 (UTC)
replies: text: D.Lazard (talk) 14:39, 23 January 2014 (UTC)
thread_title:  "not both zero" 

message: Why is this identity not considered an example of duality?  It can clearly be
formulated as duality in the optimization sense that the maximum of one set is
the minimum of another set.  Specifically, "The greatest common divisor of any
two integers is equal to the smallest positive linear combination of these
integers." As such, it has parallels not just to optimality duality.  Once can even see connections to the
duality of projective geometry where points
and lines are dual as well as fourier analysis where duality is applied between
points in the frequency domain and waves in the time domain (though I admit
this is stretching things a little, since the identity is really only
considering integer linear combinations).
It seems to me that emphasizing the dual properties of the GCD greatly
clarifies the subsequent formal development of elementary number theory, 
as well as pointing the way to more sophisticated ideas.  While this may not
be a universal perspective on Bezout's identity, I would like to know 
why it is _wrong_.  — Preceding unsigned comment added by Uscitizenjason (talk • contribs) 21:43, 29 January 2019 (UTC)
thread_title: Duality

message: The proof first claims that the remainder, r, of the division a/d of the integer, a, by the minimal element, d, of S={ax+by|x,y in Z and ax+by>0} satisfies 0 <= r < d, which is clear. But it then uses the argument that "As d is the smallest positive integer in S, the remainder r is necessarily 0, as, being a remainder, it has to be less than d". This seems incomplete to me, since the remainder could be any non-negative integer less than d - not necessarily zero. The proof given at https://proofwiki.org/wiki/B%C3%A9zout%27s_Lemma, which shows that the assumption that there is an element of S indivisible by d leads to a contradiction seems far more convincing. I also think that the proof should contain more references to published proofs or described qualitatively as a sketch, not just using a mishmash of ad-libbed logic cobbled together by different users. Jamgoodman (talk) 12:41, 28 January 2020 (UTC)
replies: text: "The remainder could be any non-negative integer in } less than d  - thus necessarily zero." (This is your sentence with my modifications in boldface). Strange that you are unable to remember the result of the preceding line. Nevertheless, I have clarified the sentence by adding some repetitions. D.Lazard (talk) 13:43, 28 January 2020 (UTC)
thread_title:  Gap in logic in the proof. We have not shown that the remainder is zero.