Diofantove jednačine — разлика између измена
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{{L|rut}}{{short description|Polinomska jednačina čija se celobrojna rešenja traže}}
'''Diofantska jednačina''' je algebarska jednačina s dve ili više nepoznatih s celobrojnim koeficijentima u kojoj se traže celobrojna ili [[Skup racionalnih brojeva|racionalna]] rešenja. Ime je dobila po [[Diofant]]u koji je prvi sistematski proučavao takve jednačine.<ref>{{Cite book|first=L. J.|last=Mordell |authorlink=Louis Mordell | title=Diophantine equations | publisher=[[Academic Press]] |year=1969 | isbn=978-0-12-506250-3 | zbl=0188.34503 | series=Pure and Applied Mathematics | volume=30 }}</ref>▼
[[File:Rtriangle.svg|thumb|Finding all [[Pythagorean triple|right triangles with integer side-lengths]] is equivalent to solving the Diophantine equation {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>}}.]]
▲'''Diofantska jednačina''' je algebarska jednačina s dve ili više nepoznatih s celobrojnim koeficijentima u kojoj se traže celobrojna ili [[Skup racionalnih brojeva|racionalna]] rešenja. Ime je dobila po [[Diofant]]u koji je prvi sistematski proučavao takve jednačine.<ref>{{Cite book|first=L. J.|last=Mordell |authorlink=Louis Mordell | title=Diophantine equations | publisher=[[Academic Press]] |year=1969 | isbn=978-0-12-506250-3 | zbl=0188.34503 | series=Pure and Applied Mathematics | volume=30 }}</ref> ''Linear Diophantine equation'' equates the sum of two or more [[monomials]], each of [[Degree of a polynomial|degree]] 1 in one of the variables, to a constant. An ''exponential Diophantine equation'' is one in which exponents on terms can be unknowns.
== Primeri ==
In the following Diophantine equations, {{math|''w''}}, {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are the unknowns and the other letters are given constants:
{| class="wikitable"
| {{math|''ax'' + ''by'' {{=}} 1}}||This is a linear Diophantine equation.
|-
| {{math|''w''<sup>3</sup> + ''x''<sup>3</sup> {{=}} ''y''<sup>3</sup> + ''z''<sup>3</sup>}}|| The smallest nontrivial solution in positive integers is 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 |accessdate=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<sup>n</sup>'' + ''y<sup>n</sup>'' {{=}} ''z<sup>n</sup>''}}||For {{math|''n''}} = 2 there are infinitely many solutions {{math|(''x'', ''y'', ''z'')}}: the [[Pythagorean triple]]s. For larger integer values of {{math|''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|authorlink=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|format=PDF|doi=10.2307/2118559|jstor=2118559|publisher=Annals of Mathematics}}</ref>) states there are no positive integer solutions {{math|(''x'', ''y'', ''z'')}}.
|-
| {{math|''x''<sup>2</sup> − ''ny''<sup>2</sup> {{=}} ±1}}|| This is [[Pell's equation]], which is named after the English mathematician [[John Pell]]. It was studied by [[Brahmagupta]] in the 7th century, as well as by Fermat in the 17th century.
|-
| {{math|{{sfrac|4|''n''}} {{=}} {{sfrac|1|''x''}} + {{sfrac|1|''y''}} + {{sfrac|1|''z''}}}}||The [[Erdős–Straus conjecture]] states that, for every positive integer {{math|''n''}} ≥ 2, there exists a solution in {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}, all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation {{math|4''xyz'' {{=}} ''yzn'' + ''xzn'' + ''xyn'' {{=}} ''n''(''yz'' + ''xz'' + ''xy'')}}.
|-
| {{math|''x''<sup>4</sup> + ''y''<sup>4</sup> + ''z''<sup>4</sup> {{=}} ''w''<sup>4</sup>}}||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.<ref>{{cite journal |author=Noam 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>
|}
== Linearne diofantske jednačine ==
Линија 209 ⟶ 229:
* {{Cite book|ref=harv|first=John | last=Stillwell | title=Mathematics and its History | edition=Second | publisher=Springer Science + Business Media Inc. |year=2004 | isbn=978-0-387-95336-6}}
* {{Cite book|ref=harv| last=Dickson | first=Leonard Eugene |author-link=Leonard Eugene Dickson | title=[[History of the Theory of Numbers]]. Volume II: Diophantine analysis | origyear=1920 | zbl=1214.11002 | mr=0245500 | location=Mineola, NY | publisher=Dover Publications |isbn=978-0-486-44233-4 |year=2005}}
* {{Cite book| first=L. J.|last=Mordell | authorlink=Louis Mordell | title=Diophantine equations | publisher=[[Academic Press]] | year=1969 | isbn=0-12-506250-8 | zbl=0188.34503 | series=Pure and Applied Mathematics | volume=30 }}
* Bashmakova, Izabella G. "Diophante et Fermat," ''Revue d'Histoire des Sciences'' 19 (1966), pp. 289-306
* Bashmakova, Izabella G. ''[[Diophantus and Diophantine Equations]]''. Moscow: Nauka 1972 [in Russian]. German translation: ''Diophant und diophantische Gleichungen''. Birkhauser, Basel/ Stuttgart, 1974. English translation: ''Diophantus and Diophantine Equations''. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
* Bashmakova, Izabella G. “[https://web.archive.org/web/20190403144351/https://core.ac.uk/download/pdf/81109310.pdf Arithmetic of Algebraic Curves from Diophantus to Poincaré]” ''Historia Mathematica'' 8 (1981), 393-416.
* Bashmakova, Izabella G., Slavutin, E.I. ''History of Diophantine Analysis from Diophantus to Fermat''. Moscow: Nauka 1984 [in Russian].
* Bashmakova, Izabella G. “Diophantine Equations and the Evolution of Algebra,” ''American Mathematical Society Translations'' 147 (2), 1990, pp. 85-100. Translated by A. Shenitzer and H. Grant.
* Rashed, Roshdi, Houzel, Christian. ''Les Arithmétiques de Diophante : Lecture historique et mathématique'', Berlin, New York : Walter de Gruyter, 2013.
* Rashed, Roshdi, ''Histoire de l’analyse diophantienne classique : D’Abū Kāmil à Fermat'', Berlin, New York : Walter de Gruyter.
{{refend}}
== Spoljašnje veze ==
{{Commonscat-lat|Diophantine equation}}▼
* -{[http://mathworld.wolfram.com/DiophantineEquation.html Diophantine Equation]. From [[MathWorld]] at [[Wolfram Research]]}-.
* {{Springer|id=d/d032610|title=Diophantine equations}}
Линија 217 ⟶ 247:
* [https://web.math.pmf.unizg.hr/nastava/metodika/materijali/diofant.pdf Diofantske jenačine]
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▲{{Commonscat|Diophantine equation}}
[[Категорија:Диофантске једначине]]
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