Solving a linear equation in a set of integers I

Abstract

(1.1) a1x1 + . . .+ akxk = b with x1, . . . , xk in a prescribed set of integers. We saw that the vanishing of the constant term b and the sum of coefficients s = a1 + . . . + ak had a strong effect on the behaviour of equation (1.1). The condition b = 0 is equivalent to homogeneity or multiplication invariance (if x1, . . . , xk is a solution, so is tx1, . . . , txk), while s = 0 means translation invariance (if x1, . . . , xk is a solution, so is x1 + t, . . . , xk+ t). We called equations with b = s = 0 invariant , and those with b 6= 0 or s 6= 0 noninvariant . In Part I of the paper we studied invariant equations; now we treat noninvariant ones. We recall the principal notations.

Locations

• Acta Arithmetica - View - PDF