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CHAPTER 13 - Some Nonlinear Diophantine Equations
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13.1 Pythagorean Triples

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Biographical information about Pythagoras can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pythagoras.html (Pythagoras)

 

Comprehensive coverage of Pythagorean triples, together with applets for finding them, can be found at http://www.math.rutgers.edu/~erowland/pythagoreantriples-project.html (Pythagorean Triples Project)

 

13.2 Fermat's Last Theorem

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An excellent survey of the history of Fermat's last theorem can be found at
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html (Fermat's last theorem)

Information about Fermat's last theorem can be found at NOVA Web site on the pages accompanying their episode devoted to Wiles's proof :
http://www.pbs.org/wgbh/nova/proof (NOVA Online - The Proof)

To begin exploring the mathematics behind Wiles's proof of Fermat's last theorem, you should look at a page developed by Charles Daney at
http://www.best.com/~cgd/home/flt/flt01.htm (The Mathematics of Fermat's Last Theorem)

Biographical information about Sophie Germain can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Germain.html (Germain)

 

 

Page 517

Biographical information about Ernst Kummer can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Kummer.html (Kummer)

 

 

Page 518

Biographical information about Andrew Wiles can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Wiles.html (Wiles)

 

 

 

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You can learn more about the Wolfskehl prize by downloading the article by Barner from the Notices of the American Mathematical Society at
http://www.ams.org/notices/199710/barner.pdf

 

 

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The companion website for the NOVA episode about Wiles’s proof of Fermat’s last theorem can be found at http://www.pbs.org/wgbh/nova/proof/ (NOVA – The Proof)

 

 

Page 522 

 

Information about Beal’s conjecture can be found at http://www.bealconjecture.com/

(Beal’s conjecture)

 

Biographical information about Eugene Catalan can be found at the MacTutor History of Mathematics Archive at

http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Catalan.html

(Catalan)

 

Background information about the Catalan conjecture can be found at http://mathworld.wolfram.com/CatalansConjecture.html (Catalan’s Conjecture)

 

To learn about the proof of Catalan’s conjecture, go to http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf (Catalan’s Conjecture:  Another Old Diophantine Problem Solved)

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Biographical information about Levi ben Gerson can be found at http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Levi.html (Levi ben Gerson)

 

 

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Background information about the Beal Conjecture, go to http://www.ams.org/notices/199711/beal.pdf (The Beal Conjecture and Prize Problem)

 

To learn more about the abc conjecture read the informative article by Granville and Tucker at http://www.ams.org/notices/200210/fea-granville.pdf (It’s as Easy As abc) or go to  http://www.math.unicaen.fr/~nitaj/abc.html (THE ABC CONJECTURE HOME PAGE)

 

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Biographical information about Bachet can be found at http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bachet.html (Bachet)

 

 

 

 

13.3 Sums of Squares

 

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You can express a positive integer of the form 4n+1 as the sum of two squares using the tool at http://www.numbertheory.org/php/serret.html

 

 

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An applet which can be used to decompose a positive integer into the sum of four squares can be found at http://www.alpertron.com.ar/FSQUARES.HTM (Sums of squares)

 

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You can find the latest results relating to Waring’s problem at http://mathworld.wolfram.com/WaringsProblem.html (Eric W. Weisstein. "Waring's Problem." From MathWorld--A Wolfram Web Resource)

 

Biographical information about Edward Waring can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Waring.html (Edward Waring) and at
http://exile.ne.mediaone.net/lucas/waring.html

 

Page 538

Biographical information about Axel Thue can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Thue.html (Thue)

 

 

13.4 Pell's Equation

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Biographical information about John Pell can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pell.html (Pell)

On-line programs for solving Pell's equation and related diophantine equations can be found at the site http://www.bioinfo.rpi.edu/~zukerm/cgi-bin/dq.html (Diophantus Quadraticus) and at

http://www.numbertheory.org/php/patz.html (Solving the diophantine equation x²-Dy²=N, D > 0 and not a perfect square)

 

 

Biographical information about Bhaskara can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Bhaskara.html (Bhaskara)