Limit The Arithmetic Properties Of The Split Function

Let n be a non-zero integer and a1<a2…<at(t≥2) be positive integers such that (a1,a2,…,at)=1.Let D(a1,a2,…,at；n) be the number of non-negative integer solutions (x1,x2,…,xt) of the Diophantine equation a1x1+a2x2+…+atxt=n. For t =2，let a=a1,b=a2 and (a,b)=1. There are a lot of results about ax by=n. T. Popoviciu [15] showed that where r=O or 1 and [x] denote the greatest integer not exceeding x. O. Bordelles [11]《Arithmetic Tales》proved that where positive integers a’，b’ satisfied that 1<a’<b,aa’≡-n(mod b);1≤b’≤ a, bb’≡-n(mod a). A. TVipathi [21] obtained that whereIn this paper we give a new proof of the formula of V(a,b;n). In particular, we obtain a triangle expression of D(a,b;n) when ab (?) n.For n = 3,let a,b,c be positive integers such that (a,b)=(b,c)=(c,a)= 1. Using partial fraction decomposition and Cauchy’s residue theorem, we get a formula of D(a,b,c;n).Especially, we obtain a triangle expression under the abc (?) n.To the general t, we also obtain the recursive relation about D(a1,a2,…,at;n).