# Diophantine Equations

Also found in: Dictionary.

## diophantine equations

[¦dī·ə¦fant·ən i′kwā·zhənz]
(mathematics)
Equations with more than one independent variable and with integer coefficients for which integer solutions are desired.

## 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 = y0an (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 = m2n2, 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 x2dy2 = 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 + … + anyn = c

(where n ≥ 3; and a0, a1, …, an and c are integers and the polynomial a0tn + a1tn−1 + … + an 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.

### REFERENCES

Gel’fond, A. O. Reshenie uravnenii v tselykh chislakh, 2nd ed. Moscow, 1956.
Dickson, L. E. History of the Theory of Numbers, vol. 2. Washington, D.C., 1920.
Skolem, Th. Diophantische Gleichungen. Berlin, 1938.
References in periodicals archive ?
The inner loop was designed using linear algebraic method via solving a set of Diophantine equation, while the outer loop was designed using LQG controller.
Test Diophantine equations Equation Solutions in the range el 4x + 9y = 91 11 e2 10x + 7y = 97 10 e3 24x + 15y = 9 13 e4 19x + 23y = 3 4 e5 [x.sup.2] + [y.sup.2] = 625 20 e6 [x.sup.2] + [y.sup.2] = 149 8 e7 [x.sup.3] + [y.sup.3] = 1008 2 e8 [x.sup.4] + [y.sup.4] = 1921 8 e9 [x.sup.5] + [y.sup.5] = 19 932 2 Table 2.
Like before, we can solve a family of systems of Diophantine equations (1).
Recursive Solution to Diophantine Equations. GPC algorithm obtained in Section 5.2 is followed by some relevant conclusions; namely, in Diophantine equations (9)-(10), [E.sub.j]([z.sup.-1]), [F.sub.j]([z.sup.-1]), [G.sub.j]([z.sup.-1]), and [H.sub.j]([z.sup.-1]) vary with predictive step number j and need recalculation.
In , Stroeker investigated the Diophantine equation
Other themes discussed during that academic year included: the Stolz--Cesaro, the intermediate and the mean value theorems, applications of complex numbers to polynomials, Diophantine equations, and geometric probability.
Ten chapters cover algebraic number theory and quadratic fields; ideal theory; binary quadratic forms; Diophantine approximation; arithmetic functions; p-adic analysis; Dirichlet characters, density, and primes in progression; applications to Diophantine equations; elliptic curves; and modular forms.
Some attempt was made by one researcher to explain this by forming and solving the associated Diophantine equations. It is noted here that potential energy and forces are determined by radial distances- that is the radii of the large circles.
In number theory many problems may be posed as diophantine equations to be solved in integers.
In [1, 2] Quadratic Diophantine equations with three unknowns have been considered for its parametric integral solutions.
In later years, students who stick with mathematics will learn more about Diophantine equations in number theory courses.
Unlike most other Diophantine equations, LDEs can be solved algorithmically (Rowe, 1986; Stewart, 1992).

Site: Follow: Share:
Open / Close