Dependence Analysis for Supercomputing pp 67-99 | Cite as

# Linear Diophantine Equations

Chapter

## Abstract

A linear diophantine equation has the form where a

$$ {a_1}{x_1} + {a_2}{x_2} + ... + {a_n}{x_n} = c $$

_{1}, a_{2},..., a_{n}, c are integer constants and x_{1}, x_{2},..., x_{n}are integer variables. As we saw in Chapter 3, these equations play a very important role in the linear dependence problem. In this chapter we study how to decide if there is a solution to a given linear diophantine equation or a system of such equations, when no further constraints are present. In either case, if a solution exists, a formula for the general solution is also obtained. A brief coverage of the greatest common divisor is given in Section 5.2, and the well-known method for solution of a single equation in two variables is described in Section 5.3. The next two sections deal with single equations in many variables and systems of equations. It is assumed that the reader is familiar with elementary properties of the greatest common divisor and basic linear algebra. For details on the gcd and Euclid’s algorithm, see the excellent treatment in [Knuth 1980, Section 4.5.2]. The matrix methods used in sections 5.4, 5.5 are taken from [Kertzner 1981]; we have changed his notation and given detailed proofs and algorithms.## Keywords

Integer Solution Dependence Analysis Diophantine Equation Great Common Divisor Elementary Matrice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Kluwer Academic Publishers 1988