Editing Diophantine equation (section)

==Examples==
In the following Diophantine equations, {{mvar|w, x, y}}, and {{mvar|z}} are the unknowns and the other letters are given constants:
{| class="wikitable"
| <math>ax+by = c</math>||This is a linear Diophantine equation, related to [[Bézout's identity]].
|-
| <math>w^3 + x^3 = y^3 + z^3</math>|| The smallest [[Triviality (mathematics)#Trivial and nontrivial solutions|nontrivial solution]] in positive integers is {{math|1=12<sup>3</sup> + 1<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup> = 1729}}.  It was famously given as an evident property of 1729, a [[taxicab number]] (also named [[Hardy–Ramanujan number]]) by [[Ramanujan]] to [[G. H. Hardy|Hardy]] while meeting in 1917.<ref>{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |title=Quotations by Hardy |publisher=Gap.dcs.st-and.ac.uk |access-date=20 November 2012 |archive-url=https://web.archive.org/web/20120716185939/http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |archive-date=16 July 2012 |url-status=dead }}</ref> There are infinitely many nontrivial solutions.<ref>{{citation|title=An Introduction to Number Theory|volume=232|series=Graduate Texts in Mathematics|first1=G.|last1=Everest|first2=Thomas|last2=Ward|publisher=Springer|year=2006|isbn=9781846280443|page=117|url=https://books.google.com/books?id=Z9MAm0lTKuEC&pg=PA117}}.</ref>
|-
| <math>x^n + y^n = z^n</math>||For {{math|1= ''n'' = 2}} there are infinitely many solutions {{math|(''x, y, z'')}}: the [[Pythagorean triple]]s. For larger integer values of {{mvar|n}}, [[Fermat's Last Theorem]] (initially claimed in 1637 by Fermat and [[Wiles's proof of Fermat's Last Theorem|proved by Andrew Wiles]] in 1995<ref name=wiles>{{cite journal|last=Wiles|first=Andrew|author-link=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://users.tpg.com.au/nanahcub/flt.pdf |journal=[[Annals of Mathematics]]|volume=141|issue=3|pages=443–551|oclc=37032255|doi=10.2307/2118559|jstor=2118559}}</ref>) states there are no positive integer solutions {{math|(''x, y, z'')}}.
|-
| <math>x^2 - ny^2 = \pm 1</math>|| This is [[Pell's equation]], which is named after the English mathematician [[John Pell (mathematician)|John Pell]]. It was studied by [[Brahmagupta]] in the 7th century, as well as by Fermat in the 17th century.
|-
| <math>\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z</math>||The [[Erdős–Straus conjecture]] states that, for every positive integer {{mvar|n}} ≥ 2, there exists a solution in {{mvar|x, y}}, and {{mvar|z}}, all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation <math>4xyz = n(yz+xz+xy).</math>
|-
| <math>x^4 + y^4 + z^4 = w^4</math>||Conjectured incorrectly by [[Euler]] to have no nontrivial solutions.  Proved by [[Elkies]] to have infinitely many nontrivial solutions, with a computer search by Frye determining the smallest nontrivial solution, {{math|1=95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup>}}.<ref>{{cite journal |authorlink=Noam Elkies |first=Noam |last=Elkies |title=On ''A''<sup>4</sup> + ''B''<sup>4</sup> + ''C''<sup>4</sup> = ''D''<sup>4</sup> |url= https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/S0025-5718-1988-0930224-9.pdf |journal=[[Mathematics of Computation]] |year=1988 |volume=51 |issue=184 |pages=825–835 |doi=10.2307/2008781 |mr=0930224 |jstor=2008781}}</ref><ref>{{cite conference|last = Frye|first = Roger E.|year = 1988|title = Proceedings of Supercomputing 88, Vol.II: Science and Applications|contribution = Finding 95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup> on the Connection Machine|doi = 10.1109/SUPERC.1988.74138|pages = 106–116}}</ref>
|}