Diophantine Equations

(named after the ancient Greek mathematician Diophantus), algebraic equations or systems of algebraic equations with integral coefficients, where the number of unknowns exceeds the number of equations and where we are looking for integral or rational solutions. The concept of Diophantine equations has been broadened in modern mathematics: they are equations in which the required solutions are algebraic numbers. Diophantine equations are also called indeterminate equations.

The simplest Diophantine equation is ax + by = 1, where a and b are relatively prime integers. This equation has an infinite number of solutions: if x₀ and y₀ are one solution, then the numbers x = x₀, + bn and y = y₀ − an (n is any integer) will also be solutions. Thus, all integral solutions of the equation 2x + 3y = 1 are obtained from the formulas x = 2 + 3n and y = −1 − 2n (here, x₀ = 2 and y₀ = −1). Another example of a Diophantine equation is x² + y² = z². The positive integral solutions of this equation are the lengths of the legs x and y and the hypotenuse z of right triangles whose sides are integral and are called Pythagorean numbers. All triplets of relatively prime Pythagorean numbers can be obtained from the formulas x = m² − n², y = 2mn, and z = m² + n², where m and n are integers (m > n > 0).

In his work Arithmetica, Diophantus attempted to find rational (not necessarily integral) solutions for special types of Diophantine equations. The general theory of Diophantine equations of the first degree was developed in the 17th century by the French mathematician C.-G. Bachet de Meziriac. By the early 19th century P. de Fermat, J. Wallis, L. Euler, J. Lagrange, and K. Gauss had investigated Diophantine equations of the type

ax² + bxy + cy² + dx + ey + f = 0

where a, b, c, d, e, and f are integers, that is, a general nonhomogeneous equation of the second degree in two unknowns. Fermat maintained, for example, that the Diophantine equation x² − dy² = 1 (the Pellian equation), where d is a positive integer that is not a perfect square, has an infinite number of solutions. Wallis and Euler provided methods for solving this equation, and Lagrange proved that the number of solutions was infinite. Using continued fractions, Lagrange studied the general nonhomogeneous Diophantine equation of the second degree in two unknowns. Gauss constructed a general theory of quadratic forms that serves as the basis for solving certain types of Diophantine equations. Only in the 20th century were important gains made in the study of Diophantine equations in two unknowns of degree higher than the second. A. Thue showed that the Diophantine equation

a₀xⁿ + a₁xⁿ⁻¹y + … + a_nyⁿ = c

(where n ≥ 3; and a₀, a₁, …, a_n and c are integers and the polynomial a₀tⁿ + a₁tⁿ⁻¹ + … + a_n is irreducible in the field of rational numbers) cannot have an infinite number of integral solutions. The British mathematician A. Baker obtained bounds on the solutions of some of these equations. The method of B. N. Delone applies to a narrower class of Diophantine equations but yields bounds on the number of solutions. In particular, his method completely solves Diophantine equations of the form

ax³ + y³ = 1

There are many directions in the theory of Diophantine equations. Thus, Fermat’s last theorem is a well-known problem in the theory of Diophantine equations. Soviet mathematicians (B. N. Delone, A. O. Gel’fond, and D. K. Faddeev, among others) have made fundamental contributions to the theory of Diophantine equations.