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11D: Diophantine equations


Introduction

History

Applications and related fields

See also 11GXX, 14GXX. In particular, discussion of many examples and families of equations has been moved to pages for (arithmetic) algebraic geometry; the dividing line is unclear -- sorry.

Waring's problem and its ilk are considered 11P: Additive Number Theory, as are representations as sums of squares and so on. (Thus the Diophantine equation x^2+y^2=N can be treated both in 11P and here in 11D (as a Pell equation).)

Some Diophantine equations are best thought of as part of 11J: transcendental number theory. For example, Catalan's conjecture (8 and 9 the only consecutive powers) and many others with unknown integer exponents are part of that area. [Schematic of subareas and related areas]

Subfields

Parent field: 11 - Number Theory

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

Apart from texts focusing broadly on Number Theory or narrowly on, say, Fermat's Last Theorem (which is treated in section 11D41) there are comparatively few texts with focus in this area.

There is a nice short survey article by Beukers, F.; Manin, Yu. I.: "Diophantine equations", Nieuw Arch. Wisk. (4) 7 (1989), 3--13.

Faisant, Alain: "Résolution de l'équation du second degré en nombres entiers", Séminaire d'Analyse, 1987--1988, Exp. No. 23, 15 pp., Univ. Clermont-Ferrand II, Clermont-Ferrand, 1990. -- a thorough summary of the case of integral binary quadratic equations.

See also the references for number theory in general.

Software and tables

Other web sites with this focus

Selected topics at this site


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Last modified 2000/01/14 by Dave Rusin. Mail: