| Professor F.Smarandache is a famous Rumanian number-theoretic expert. One of his numerous contributions is the excellent unsolved problems that were presented by him continually.In 1993,Professor F.Smarandache presented 105 unsolved problems in Only Problems,Not Solutions , it arose great interests for scholars.In this paper. I study the 23th. 57th, 65th, and 83th problems, and give several interesting asymptotic formulae.Professor F.Smarandache asked us to find the maximal number r such that : the set (1,2, ...,r} can be partitioned into n classes such that no class contains integers x,y,z with xy =z.In chapter 2 ,1 give a low bound estimate for r,and prove that r > n9.In chapter 3, 4 and 5 ,I study some properties of simple number,m-power residues, and the integer part of k-power root of positive integer.For any positive integer n, a number n is called simple number if the product of its proper divisors is less than or equal to n; a(n) is the number n released of its m-power residues: if n = p1a1.. . pTar, with all pi primes and all ai>1, then a(n) = p1b1...prbr,bi = min(m - 1,ai); mq(n)is the superior integer part of k-power root of n,then mq(n) = [n1/k].The main purpose of this paper is to study:where A is a set of simple numbers;where a(n) is the number n released of its m-power residues;where mq(n)is the superior integer part of k-power root of n. Several regular results and several interesting asymptotic formulae are obtained. |
