An inequality between prime powers dividing n !

For any positive integer n 2: 1 and for any prime number p let ep(n) be the exponent at which the prime p appears in the prime factor decomposition of nL In this note we prove the following; Theorem. Let p < q be two prime numbers, and let n > 1 be a positive integer such that pq I n. Then, (1) Inequality (1) was suggested by Balacenoiu at the First International Conference on Smarandache ::\otions in Number Theory (see [1]). In fact, in [1], Balacenoiu showed that (1) holds for p = 2. In what follows we assume that p 2: 3. \Ve begin with the following lemmas: Lemma 1. (i) The function is increasing for x 2: e. x-I j(x) =-log x (ii) Let p 2: 3 be a real number. Then, for x 2: p. (iii) Let p 2: 3 be a real number. The function x-2 is positive and decreasing for x 2: p(p + 2). (iv) p+2 log(p+4)p logp (v) p + 1 > log(p + 2) p logp for p > e.