[Search][Subject Index][MathMap][Tour][Help!]

	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

• 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
• 11D72: Equations in many variables, See also 11P55
• 11D75: Diophantine inequalities, See also 11J25
• 11D79: Congruences in many variables
• 11D85: Representation problems, See also 11P55
• 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.

See also the references for number theory in general.

Software and tables

Other web sites with this focus

• An online calculator which solves binary quadratic equations integrally; see also the adjoining page outlining the methods of solution.

Selected topics at this site

• Difficulties writing a program to "solve" Diophantine equations
• An exercise in viewing one equation from many perspectives: x^2 + 7 =8 p^n with p prime
• What is the method of infinite descent?
• The Tarry-Escott multigrades problem: given a positive integer n, find two sets of integers a_1, ..., a_r and b_1, ..., b_r, with r as small as possible, such that sum (a_j)^k = sum (b_j)^k for k = 1, 2, ..., n. Conjecture: r=n+1 for all n.
• The multigrades problem (find sets of integers whose sums are equal, sums of squares, sums of cubes,...)
• New solution of the Prouhet-Tarry-Escott problem for k=11, other limitations
• Suggested by an arrangement of numbers in a basketball tournament: solve ab = c + d, cd = a + b in integers.
• The Times puzzle: find rational solutions to x^3+y^3=6. (an elliptic curve)
• Questions related to an Erdös conjecture: that 4/n = 1/x + 1/y + 1/z has a solution for every natural number n.
• Solutions to a^6 + 5(a^4)b + 6(a^2)(b^2) + b^3 = 1 in integers
• Generate all (small) Pythagorean triples
• Integer triangles with one angle measuring 120 degrees
• Runge's theorem giving a constructive bound on the number of solutions to certain Diophantine equations in two variables
• A pair of equations becomes a single equation over the Gaussian integers
• When can a 2-variable quadratic equation be solved in integers?
• Finding all integer solutions to a quadratic equation in two variables
• Solving Pell's equation x^2+dy^2=N ( esp: N \not= 1 ).
• Citation for solving Pell's equation (for N \not= 1)
• For which D is x^2 - D y^2 = +/-4 solvable?
• For which D and N is x^2 - D y^2 = N solvable?
• Various methods to solve Pell's equations, with citations, special cases, etc.
• Long summary: triangular numbers which are perfect squares and related topics
• Which triangular numbers are squares? (example of Pell's equation).
• Solution to X^2 - D Y^2 = 1 in polynomials D,X,Y
• Solving x^2+xy+y^2=z^2 to make nice calculus problems.
• Among solutions of 3 x^2 + 5 y^2 = 2^(2n+1), estimate growth of min(x,y).
• Finding all integral solutions to a homogeneous quadratic in 3 variables -- example.
• [Offsite] Chen Shuwen has a web page covering the whole range of problems of equal sums of like powers.
• Near misses of the Fermat equation.
• Lit review and pointer for equations x^n+y^n=2*z^n [Ken Ribet]
• Finding solutions to a single multivariable homogeneous quadratic equation
• Parameterizing the solution set to a quadratic
• Thue equations (homogeneous 2-variable polynomial= const)
• Solve x^n + d y^n = c: Thue equations.
• Pointer for exponential Diophantine equations (e.g. 3^x+5^y=y^z+1 ).
• Find integers with each xi*xj + 1 a square (forced, if (xy+1)(xz+1)(yz+1) is square)
• Discussion of triangles whose sides are of rational length.