On the Smarandache function and the Fermat number

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Scientia Magna Vol. 4 (2008), No. 2, 25-28

On the Smarandache function and the Fermat number Jinrui Wang Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China Abstract For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n | m!. That is, S(n) = min{m : n | m!, n ∈ N }. The main purpose of this paper is using the elementary method to study the estimate problem n of S (Fn ), and give a sharper lower bound estimate for it, where Fn = 22 + 1 is called the Fermat number. Keywords F. Smarandache function, the Fermat number, lower bound estimate, elementary method.

§1. Introduction and result For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n | m!. That is, S(n) = min{m : n | m!, n ∈ N }. For example, the first few values of S(n) are S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 11, S(12) = 4, · · · . About the elementary properties of S(n), many authors had studied it, and obtained some interesting results, see references [1], [2], [3], [4] and [5]. For example, Lu Yaming [2] studied the solutions of an equation involving the F.Smarandache function S(n), and proved that for any positive integer k ≥ 2, the equation S(m1 + m2 + · · · + mk ) = S(m1 ) + S(m2 ) + · · · + S(mk ) has infinite group positive integer solutions (m1 , m2 , · · · , mk ). Dr. Xu Zhefeng [3] studied the value distribution problem of S(n), and proved the following conclusion: Let P (n) denotes the largest prime factor of n, then for any real number x > 1, we have the asymptotic formula à 3 ! ¡ ¢ 3 X 2ζ 32 x 2 x2 2 (S(n) − P (n)) = , +O 3 ln x ln2 x n≤x where ζ(s) denotes the Riemann zeta-function. Chen Guohui [4] studied the solvability of the equation S 2 (x) − 5S(x) + p = x,

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