关于线性型Frobenius问题的注记

A NOTE ON FROBENIUS PROBLEM OF LINEAR FORMS

以g(a_1,a_2,…,a_n)表n元整系数线性型a_1x_1+…+a_nx_n,a_i>0,(a_1,…,a_n)=1,不可非负整表出之最大整数,D_(n-1)=(a_1,…,a_(n-1)).注记中将证明g(a_1,…,a_n)=D_(n-1)·g(a_1/D_(n-1),…,a_(n-1)/D_(n-1),a_n)+(D_(n-1)-1)a_n。并由此对Brayer关于g(a_1,…,a_n)之上确界的著名结果和Roberts关于g(a,a+d,…,a+sd)的精确结果分别给出一个十分简洁的新证明.

Let g (a1…,an) be the greatest integer N for which the Diophantine equation a1xi=N has no solution in non-negative integers, where (a1,…,an)=1.In this note, the following result is proved: g(a1,…,an)=Dn-1(Dn-1-1)an,where Dn-1=gcd (a1,…,an-1).From this, we provide a very simple proof for the Brauer's result and Roberts's result respectively.