关于n元线性方程正整数解的个数

On the Number of Positive Integer Solutions of n-Element Linear Equations

设ai,b(i=1 ,2 ,… ,n)∈N ,gcd(a1,a2 ,… ,an) =1 ,给出n元线性方程a1x1+a2 x2 +…+anxn) =b正整数解个数的递推公式为m =∑ll=1 ki,其中l=[(b -a1-… -an- 1) /an],ki 表示方程∑n - 1j=1ajxj=b-ian

Let \%b,a\-i(i=1,2,…,n)\%be positive integers with gcd(\%a\-1,a\-2,…,a\-n)=1\%,a formula about the number of positive integer solutions(\%x\-1,x\-2,…,x\-n\%) on equation \%a\-1x\-1+a\-2x\-2+…+a\-nx\-n=b\%is given as follows:\%m=∑li=1k\-i\%.here,\%l=\[(b-a\-1-a\-2-…-a\-\{n-1\})/a\-n\%\],\%k\-i\%:the number of positive integers solutions on \%a\-1x\-1+a\-2x\-2+…+a\-\{n-1\}x\-\{n-1\}=b-ia\-n\%.