Vol. 39, No. 1, 1971

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On the solution of linear G.C.D. equations

David Jacobson and Kenneth S. Williams

Vol. 39 (1971), No. 1, 187–206
Abstract

Let Z denote the domain of ordinary integers and let

m(≧ 1),n(≧ 1),li (i = 1,⋅⋅⋅ ,m),lij(i = 1,⋅⋅⋅ ,m; j = 1,⋅⋅⋅ ,n) ∈ Z.

We consider the solutions x Zn of G.C.D.

(1)  (l11x1 + ⋅⋅⋅+ l1nxn + l1,⋅⋅⋅ ,lm1x1 + ⋅⋅⋅+ lmnxn + lm,c) = d,

where c(0),d(1) Z and G.C.D. denotes “greatest common divisor”. Necessary and sufficient conditions for solvability are proved. An integer t is called a solution modulus if whenever x is a solution of (1), x + tg is also a solution of (1) for all y Zn. The positive generator of the ideal in Z of all such solution moduli is called the minimum modulus of (1). This minimum modulus is calculated and the number of solutions modulo it is derived.

Mathematical Subject Classification
Primary: 10C99
Milestones
Received: 30 November 1970
Revised: 1 April 1971
Published: 1 October 1971
Authors
David Jacobson
Kenneth S. Williams