# On the indexes of beauty.

Abstract For any fixed positive integer m, if there exists a positive integer n such that n = m x d(n), then m is called an index of beauty, where d(n) is the Dirichlet divisor function. In this paper, we shall prove that there exist infinite positive integers such that each of them is not an index of beauty.

Keywords Dirichlet divisor function, index of beauty, elementary method.

[section] 1. Introduction and result

For any positive integer n, the famous Dirichlet divisor function d(n) is defined as the number of all distinct divisors of n. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the prime power factorization of n, then from the definition and properties of d(n) we can easily get

d(n) = ([[alpha].sub.1] + 1) x ([[alpha].sub.2] + 1) ... ([[alpha].sub.k] + 1). (1)

About the deeply properties of d(n), many people had studied it, and obtained a series results, see references [2], [3] and [4]. In reference [5], Murthy introduced an index of beauty involving function d(n) as follows: For any positive integer m, if there exists a positive integer n such that

m = n/d(n),

then m is called an index of beauty. At the same time, he also proposed the following conjecture:

Conjecture. Every positive integer is an index of beauty.

Maohua Le [6] gave a counter-example, and proved that 64 is not an index of beauty. We think that the conclusion in [6] can be generalization. This paper as a note of [6], we shall use the elementary method to prove the following general conclusion:

Theorem. There exists a set A including infinite positive integers such that each of n [member of] A is not an index of beauty.

[section] 2. Proof of the theorem

In this section, we shall prove our Theorem directly. For any prime p [greater than or equal to] 5, if we taking m = [p.sup.p-1], then we can prove that m is not an index of beauty. In fact, if m is an index of beauty, then there exists a positive integer n such that n = [p.sup.p-1] x d(n). From this identity we can deduce that [p.sup.p-1] | n. Let n = [p.sup.[alpha]] x b and (p, b) = 1, then from n = [p.sup.p-1] x d(n) and (1) we have the identity

[p.sup.[alpha]-p+1] x b = ([alpha] + 1) x d(b). (2)

From (2) we may immediately deduce that [alpha] [greater than or equal to] p-1. If [alpha] = p - 1, then (2) become b = p x d(b), this contradiction with (b, p) = 1. If [alpha] = p, then (2) become p x b = (p + 1) d(b), or

b = (1 + 1/p) x d(b). (3)

It is clear that (3) is not possible if prime p [greater than or equal to] 5. In fact if b = 1, 2, 3, 4, 5, 6, 7, then it is easily to check that (3) does not hold. If b [greater than or equal to] 8, then from (1) and the properties of d(b) we can deduce that b [greater than or equal to] 3/2 x d(b) > (1 + 1/p) x d(b). So (3) does not also hold. So if [alpha] = p, then (2) is not possible. If [alpha] [greater than or equal to] p + 1, let [alpha] - p = k [greater than or equal to] 1, the (2) become

[p.sup.k+1] x b = (p + k + 1) x d(b). (4)

Note that b [greater than or equal to] d(b) for all positive integer b, so formula (4) is not possible, since [p.sup.k+1] > p + k + 1 for all prime p [greater than or equal to] 5 and k [greater than or equal to] 1. Since there are infinite prime p [greater than or equal to] 5, and each [p.sup.p-1] is not an index of beauty, so there exists a set A including infinite positive integers such that each of n [member of] A is not an index of beauty. This completes the proof of Theorem.

References

[1] F.Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 1993.

[2] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem (in Chinese), Shanghai Science and Technology Press, Shanghai, 1988.

[3] Tom M. Apostol, Introduction to Analytical Number Theory, Spring-Verlag, New York, 1976.

[4] Aleksandar Ivie, The Riemann Zeta-Function, Dover Publications, New York, 2003.

[5] A. Murthy, Some more conjectures on primes and divisors, Smarandache Notions Journal, 12(2001), 311-312.

[6] Mohua Le, A conjecture concerning indexes of beauty, Smarandache Notions Journal, 14(2004), 343-345.

[7] M. L. Perez, Florentin Smarandache, definitions, solved and unsolved problems, conjectures and theorems in number theory and geometry, Xiquan Publishing House, 2000.

[8] Kenichiro Kashihara, Comments and topics on Smarandache notions and problems, Erhus University Press, USA, 1996.

[9] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi'an, 2007.

[10] Luo Genghua, Introduction to number theory, Science Press, Beijing, 1958.

[11] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford University Press, Oxford, 1937.

Hongyan Liu

Department of Applied Mathematics, Xi'an University of Technology, Xi'an, P.R.China

(1) This work is supported by the Shaann Provincial Education Department Foundation 08JK398.