Indeterminate Equation.

Indeterminate Equation. Indeterminate Equation, in mathematics, an equation that has no set of unique solutions and that therefore cannot be uniquely solved. An indeterminate equation may have an infinite number of solutions. Such equations become solvable in the ordinary sense only when additional restrictions are put on the problem; a common restriction is that the solutions must be expressed in integers. A simple example of this type of problem is the following: In what ways can change be given for a half-dollar in nickels and quarters? Algebraically this reduces to the equation 5x + 25y = 50. This equation has an infinite number of solutions if fractional solutions are admissible, but the phrasing of the problem prohibits such solutions because a third of a nickel, for example, is meaningless. With this restriction, it is apparent that three and only three solutions exist: ten nickels and no quarters, five nickels and one quarter, or no nickels and two quarters. Some such problems have no solutions, for example, the following one: In what ways can change be given for 37 cents in nickels and quarters? In more complex problems, the solution or solutions are not apparent, and an extensive algebra has been developed to determine the solutions. The simplest such problems can be expressed in the form of a linear algebraic equation in two unknowns (such as the equation shown above) and can be solved by a method developed by the ancient Greek mathematicians Diophantus and Euclid. The solutions, if any exist, are determined by finding the largest common divisor of the numbers by which x and y are multiplied in the equation; in the equation shown above, the numbers are 5 and 25, and their largest common divisor is 5. If this largest common divisor divides the number on the right side of the equation (as 5 divides 50), the equation has one or more integral solutions. For some further simple examples of indeterminate problems, see Diophantine Analysis. Many of the greatest mathematicians, such as Carl Friedrich Gauss, a German, spent much time developing integer solutions of exceedingly complex indeterminate equations. See also Fermat's Last Theorem. Contributed By: James Singer Reviewed By: J. Lennart Berggren Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.