Smarandache Notions Journal, Vol. 9, No.1-2-3, 1998, pp. 124-125. Maoh:.::.a Le Zhanjiang Normal College, Zhanjang, Guangdong, ?R.China Abstract. For any positive integer n, let S(n) denote the S~arandache function of n. In this paper Ne prove that S(m)+S(n). S(~n)~ Let N be the set of a~~ posicive integers. For any posicive integer n, let S(n) denote the Smarandache function of n. By ~2~, '.f'le ::ave : -: \ S(nj= min{~!kE N, Dik!}. \ - I ~ecent:y, Jozsef[l] proved that (2) S(mn)~ rnS(n), m,n E N. In this paper we give a considerable improvement for the ".lpper bound (2). We prove the following result. Theorem. For any positive integers m,n, we have S(m)+S(n). S(rnn)~ Proof. a=S(m) and b=5(n) Then we have ~et ,.,Ib' l ... j :, ~et x be a positive integer with x ~ 0, and :et x (x-I) a! x be a binomial coeficient. It is a wel: k~own fact that a a positive integer. 50 we r-:ave ~s 1,J-;' a ;' :: _x (x-1.J. . I\ ••• (\ x-a·T 1_ \/ , by ! 4- \ . ::..:rther, since m!a!, we get from (5) , / (6 ; mix(x-1) ... (x-a+l), fer any positive integer x Nith x ~ a. ?u~ x=a+b. We see from (3) and (6) that (i \ I. ! i ?ererences 1. S.Jozsef,On certain inegualites involving t~e Smarandac~e function, Smarandache No~ce J. 7(l996), No.1-3, 3-6. 2. ? Smarandache "A function in the nurrber t.heory"," Smarandac:-.e fU:1ction J". 1(1990), No.1, 3-17. 125