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# 11D: Diophantine equations

## 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.

• Diophantine equations whose solution set is one-dimensional are discussed with algebraic curves. This includes single equations in 2 variables (or homogeneous equations in 3 variables, such as the Fermat equation). In particular,...
• Equations whose solutions are curves of genus 1 are discussed in the subsection on elliptic curves. Examples include cubics in two variables, homogeneous cubics in three variables, pairs of quadratics in four variables, and equations of the form y^2=Q(x) where Q is a polynomial of degree 3 or 4.
• Sets of N equations in N+2 variables (or N+3 variables, if those equations are homogeneous) describe algebraic surfaces; for example the question of the existence of a "rational box" is there.

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.

## Subfields

• 11D04: Linear equations
• 11D09: Quadratic and bilinear equations
• 11D25: Cubic and quartic equations
• 11D41: Higher degree equations; Fermat's equation
• 11D45: Counting solutions of Diophantine equations [new in 2000]
• 11D57: Multiplicative and norm form equations
• 11D59: Thue-Mahler equations [new in 2000]
• 11D61: Exponential equations
• 11D68: Rational numbers as sums of fractions
• 11D79: Congruences in many variables
• 11D88: p-adic and power series fields
• 11D99: None of the above but in this section

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.

• Mordell, L. J.: "Diophantine equations", Academic Press, London-New York 1969, 312 pp. -- a recommended overview.
• Lang, Serge: "Fundamentals of Diophantine geometry", Springer-Verlag, New York-Berlin, 1983.370 pp. ISBN 0-387-90837-4
• Sprindzuk, Vladimir G., "Classical Diophantine equations", Lecture Notes in Mathematics, 1559. Springer-Verlag, Berlin, 1993. 228 pp. ISBN 3-540-57359-3

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.