Editing Pell's equation (section)

== Generalized Pell's equation ==
The equation
<math display="block">x^2 - n y^2 = N</math>
is called the '''generalized'''<ref name="Peker_2021">{{cite book |last=Peker |first=Bilge |date=2021 |title=Current Studies in Basic Sciences, Engineering and Technology 2021 |url=https://www.isres.org/books/chapters/CSBET2021_10_03-01-2022.pdf |publisher= ISRES Publishing |page=136 |access-date=2024-02-25}}</ref><ref name="Tamang_2022">{{cite report |last=Tamang |first=Bal Bahadur |date=August 2022 |title=Solvability of Generalized Pell's Equation and Its Applications in Real Life |url=https://www.researchgate.net/publication/363254319 |publisher=Tribhuvan University |access-date=2024-02-25}}</ref> (or '''general'''<ref name="titu">{{Cite book |last1=Andreescu |first1=Titu |title=Quadratic Diophantine Equations |last2=Andrica |first2=Dorin |publisher=Springer |year=2015 |isbn=978-0-387-35156-8 |location=[[New York City|New York]]}}</ref>) '''Pell's equation'''. The equation <math>u^2 - n v^2 = 1</math> is the corresponding '''Pell's resolvent'''.<ref name="titu"/> A recursive algorithm was given by Lagrange in 1768 for solving the equation, reducing the problem to the case <math>|N| < \sqrt{n}</math>.<ref>{{Cite book |last=Lagrange |first=Joseph-Louis |url=https://gallica.bnf.fr/ark:/12148/bpt6k215570z |title=Oeuvres de Lagrange. T. 2 / publiées par les soins de M. J.-A. Serret [et G. Darboux]; [précédé d'une notice sur la vie et les ouvrages de J.-L. Lagrange, par M. Delambre] |date=1867–1892 |language=fr}}</ref><ref>{{Cite web |last=Matthews |first=Keith |title=The Diophantine Equation ''x''<sup>2</sup> − ''Dy''<sup>2</sup> = ''N'', ''D'' > 0 |url=http://www.numbertheory.org/pdfs/patz5.pdf |archive-url=https://web.archive.org/web/20150318090657/http://www.numbertheory.org/pdfs/patz5.pdf |archive-date=2015-03-18 |url-status=live |access-date=20 July 2020}}</ref> Such solutions can be derived using the continued-fractions method as outlined above.

If <math>(x_0, y_0)</math> is a solution to <math> x^2 - n y^2 = N,</math> and <math>(u_k, v_k)</math> is a solution to <math>u^2 - n v^2 = 1,</math> then <math>(x_k, y_k)</math> such that <math>x_k + y_k \sqrt{n} = \big(x_0 + y_0 \sqrt{n}\big)\big(u_k + v_k \sqrt{n}\big)</math> is a solution to <math>x^2 - n y^2 = N</math>, a principle named the ''multiplicative principle''.<ref name="titu"/> The solution <math>(x_k, y_k)</math> is called a ''Pell multiple'' of the solution <math>(x_0, y_0)</math>.

There exists a finite set of solutions to <math>x^2 - n y^2 = N</math> such that every solution is a Pell multiple of a solution from that set. In particular, if <math>(u, v)</math> is the fundamental solution to <math>u^2 - n v^2 = 1</math>, then each solution to the equation is a Pell multiple of a solution <math>(x, y)</math> with <math>|x| \le \tfrac{1}{2} \sqrt{|N|} \left(\sqrt{|U|} + 1\right)</math> and <math>|y| \le \tfrac{1}{2 \sqrt{n}} \sqrt{|N|} \left(\sqrt{|U|} + 1\right) </math>, where <math>U = u + v \sqrt n</math>.<ref name="kconrad">{{cite web |last1=Conrad |first1=Keith |title=PELL'S EQUATION, II |url=https://kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn2.pdf |access-date=14 October 2021}}</ref>

If ''x'' and ''y'' are positive integer solutions to the Pell's equation with <math>|N| < \sqrt n</math>, then <math>x/y</math> is a convergent to the continued fraction of <math>\sqrt n</math>.<ref name="kconrad" />

Solutions to the generalized Pell's equation are used for solving certain [[Diophantine equation]]s and [[unit (ring theory)|units]] of certain [[ring (mathematics)|ring]]s,<ref>{{Cite journal |last=Bernstein |first=Leon |date=1975-10-01 |title=Truncated units in infinitely many algebraic number fields of degreen ≧4 |journal=Mathematische Annalen |language=en |volume=213 |issue=3 |pages=275–279 |doi=10.1007/BF01350876 |s2cid=121165073 |issn=1432-1807}}</ref><ref>{{Cite journal |last=Bernstein |first=Leon |date=1 March 1974 |title=On the Diophantine Equation ''x''(''x'' + ''n'')(''x'' + 2''n'') + ''y''(''y'' + ''n'')(''y'' + 2''n'') = ''z''(''z'' + ''n'')(''z'' + 2''n'') |journal=Canadian Mathematical Bulletin |language=en |volume=17 |issue=1 |pages=27–34 |doi=10.4153/CMB-1974-005-5 |s2cid=125002637 |issn=0008-4395|doi-access=free }}</ref> and they arise in the study of [[SIC-POVM]]s in [[quantum information theory]].<ref>{{Cite journal |last1=Appleby |first1=Marcus |last2=Flammia |first2=Steven |last3=McConnell |first3=Gary |last4=Yard |first4=Jon |date=August 2017 |title=SICs and Algebraic Number Theory |journal=[[Foundations of Physics]] |language=en |volume=47 |issue=8 |pages=1042–1059 |arxiv=1701.05200 |bibcode=2017FoPh...47.1042A |doi=10.1007/s10701-017-0090-7 |s2cid=119334103 |issn=0015-9018}}</ref>

The equation
<math display="block">x^2 - n y^2 = 4</math>
is similar to the resolvent <math>x^2 - n y^2 = 1</math> in that if a minimal solution to <math>x^2 - n y^2 = 4</math> can be found, then all solutions of the equation can be generated in a similar manner to the case <math>N = 1</math>. For certain <math>n</math>, solutions to <math>x^2 - n y^2 = 1</math> can be generated from those with <math>x^2 - n y^2 = 4</math>, in that if <math>n \equiv 5 \pmod{8},</math> then every third solution to <math>x^2 - n y^2 = 4</math> has <math>x, y</math> even, generating a solution to <math>x^2 - n y^2 = 1</math>.<ref name="titu" />