Linear Diophantine Equations

  • Utpal Banerjee
Part of the The Kluwer International Series in Engineering and Computer Science book series (SECS, volume 60)


A linear diophantine equation has the form
$$ {a_1}{x_1} + {a_2}{x_2} + ... + {a_n}{x_n} = c $$
where a1, a2,..., an, c are integer constants and x1, x2,..., xn 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.


Integer Solution Dependence Analysis Diophantine Equation Great Common Divisor Elementary Matrice 
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Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • Utpal Banerjee
    • 1
  1. 1.Control Data CorporationSunnyvaleUSA

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