Talk:Diophantine equation

message: Forty years ago I discovered that the Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc) can be generated from the second degree Diophantine equation
     5k^2 -/+ 4 = m^2    where the -,+ is taken alternately.
The equation is a variation on Pell's, in that
     x^2 - ny^2 = +/-  4   instead of 1.
Walter Abetz 13:32, 19 November 2006 (UTC)
thread_title:  Fibonacci Sequence and its Second Degree Diophantine Equation 

message: I'm personally not clear on the distinction between Arithmetic Geometry and Algebraic Geometry, except some vague awareness that the former usually deals with rational points on varieties and the latter with more general algebraic structure.  So I can kind of see why 'Arithmetic geometry' redirects to this page, since finding rational points on varieties is roughly speaking the same problem as solving Diophantine equations.  But it'd be nice if someone who knows something about the subject could insert a paragraph or two on this page about it, if not create an actual page for it. Chenxlee (talk) 17:55, 11 February 2008 (UTC)
thread_title:  Arithmetic Geometry 

message: In the beginning of the article a Diophantine equation is defined to be a _polynomial_ equation (with certain properties and conditions), but one of the given examples, namely 4/n = 1/x + 1/y + 1/z, is in fact not a polynomial equation by the definition given in the article about "polynomial" because it contains division.
Please someone fix that.
Either change the definiton of an Diophantine equation or remove the offending example. (I dont know which is the correct way)  —Preceding unsigned comment added by 213.240.234.31 (talk) 19:58, 11 November 2008 (UTC)
replies: text: The example is trivially modifiable to be a polynomial: 4xyz = yzn + xzn + xyn. So it's not really a violation of the definition, but maybe some explanation of why it isn't might be useful. —David Eppstein (talk) 20:03, 11 November 2008 (UTC)
replies: text: Thank you. (Now i'm happy:))  —Preceding unsigned comment added by 213.240.234.31 (talk) 20:14, 11 November 2008 (UTC)
thread_title:  Inconsistency in the article 

message: "...[a polynomial equation] that allows two or more variables to take integer values only."
Isn't this saying the same thing as "...that restricts two or more variable to integer values only", and if so, isn't the second wording clearer?
74.192.47.171 (talk) 05:21, 25 December 2013 (UTC)
replies: text: — D.Lazard (talk) 12:02, 25 December 2013 (UTC)
thread_title:  Definition reads oddly 

message: I think the article should mention how to expand the solution from two variables to more. Here is a helpful link:
https://math.stackexchange.com/questions/20906/how-to-find-an-integer-solution-for-general-diophantine-equation-ax-by-cz  — Preceding unsigned comment added by W3ricardo (talk • contribs) 04:29, 15 April 2018 (UTC)
replies: text: The general case of linear Diophantine equations (any number of unknowns, any number of equations) is treated two subsections later. D.Lazard (talk) 08:52, 15 April 2018 (UTC)
thread_title:  One linear equation with more than two variables 

message: even if the variables were all simply one, wouldn't this be infinite?  unless zeros are allowed. --142.163.195.48 (talk) 20:46, 26 January 2021 (UTC)
thread_title:  infinite equation 

message: Should Sums of three cubes be included in the 'see also' section?  somewhat popular among the numberphile crowd. --142.163.195.48 (talk) 20:58, 26 January 2021 (UTC)
thread_title:  "Sums of three cubes" 

message: Maybe we should have axy + bx + cy = d in the examples of diophantine equations. I believe it's is a good example because it is both common and easy to solve—using SFFT ("Simon's Favorite Factoring Trick"):
\begin{align}
cxy + ax + by & = d \\ 
\Longleftrightarrow (cx + b)(cy + a) & = ab + cd
\end{align}
Now you can consider the factors of ab + cd to solve for (x, y). Revol Ufiaw (talk) 09:55, 29 May 2024 (UTC)
thread_title:  Simon's Favorite Factoring Trick