On a generalized equation of Smarandache and its integer solutions

Scientia Magna Vol. 2 (2006), No. 2, 82-84

On a generalized equation of Smarandache and its integer solutions Chuan Lv School of Mathematics and Computational Science, China University of Petroleum Dongying Shandong, P.R.China Abstract Let a 6= 0 be any given real number. If the variables x1 , x2 , · · · , xn satisfy x1 x2 · · · xn = 1, the equation 1 x2 1 xn 1 x1 a + a + ··· + a = na x1 x2 xn has one and only one nonnegative real number solution x1 = x2 = · · · = xn = 1. This generalized the problem of Smarandache in book [1].

Keywords

Equation of Smarandache, real number solutions.

§1. Introduction Let Q denotes the set of all rational numbers, a ∈ Q \ {−1, 0, 1}. In problem 50 of book [1], Professor F. Smarandache asked us to solve the equation 1

xa x +

1 x a = 2a. x

(1)

Professor Zhang [2] has proved that the equation has one and only one real number solution x = 1. In this paper, we generalize the equation (1) to 1 x1 1 1 xn a + ax2 + · · · + a = na, x1 x2 xn

(2)

and use the elementary method and analysis method to prove the following conclusion: Theorem. For any given real number a 6= 0, if the variables x1 , x2 , · · · , xn satisfy x1 x2 · · · xn = 1, then the equation 1 x1 1 1 xn a + ax2 + · · · + a = na x1 x2 xn has one and only one nonnegative real number solution x1 = x2 = · · · = xn = 1.

§2. Proof of the theorem In this section, we discuss it in two cases a > 0 and a < 0.

1) For the case a > 0, we let f (x1 , x2 , · · · , xn−1 , xn ) =

1 x1 1 1 xn−1 1 xn + a + ax2 + · · · + a a − na, x1 x2 xn−1 xn

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On a generalized equation of Smarandache and its integer solutions

83

If we take xn as the function of the variables x1 , x2 , · · · , xn−1 , we have f (x1 , x2 , · · · , xn−1 , xn ) =

1 1 1 xn−1 1 x1 a + ax2 + · · · + a + x1 x2 · · · xn−1 a x1 x2 ···xn−1 − na. x1 x2 xn−1

Then the partial differential of f for every xi (i = 1, 2, · · · , n − 1) is µ ¶ 1 ∂f 1 xi 1 1 = a log a − + a x1 x2 ···xn−1 (x1 x2 · · · xn−1 − log a) ∂xi xi xi xi µ µ ¶ µ ¶¶ 1 1 1 = axi log a − + axn − log a . xi xi xn Let µ xi

g(x1 , x2 , · · · , xn−1 , xn ) = a

the partial differential quotient of g is µ ∂g log a 1 xi = a log2 a − + 2 ∂xi xi xi õ ¶2 1 axi x log a − + = i x2i 2

1 log a − xi

¶

µ xn

+a

¶ 1 − log a , xn

(3)

¶ ¢ axn ¡ 2 2 + x log a − xn log a + 1 xi xn n ! ! õ ¶2 3 axn 3 1 + + > 0. xn log a − 4 xi xn 2 4

It’s easy to prove that the function u(x) = ax (log a − x1 ) is increasing for the variable x when x > 0. From (3) we have: ∂f i) if xi > xn , g > 0, ∂x > 0, and f is increasing for the variable xi ; i ∂f < 0, and f is decreasing for the variable xi ; ii) if xi < xn , g < 0, ∂x i ∂f iii) if xi = xn , g = 0, ∂xi = 0, and we get the minimum value of f .

We have f ≥ fx1 =xn ≥ fx1 =x2 =xn ≥ · · · ≥ fx1 =x2 =···=xn ≥ fx1 =x2 =···=xn =1 = 0, and we prove that the equation (2) has only one integer solution x1 = x2 = · · · = xn = 1.

2) For the case a < 0, the equation (2) can be written as 1 1 1 (−1)x1 |a|x1 + (−1)x2 |a|x2 + · · · + (−1)xn |a|xn = −n|a|, x1 x2 xn

(4)

so we know that xi (i = 1, 2, · · · , n) is not an irrational number.

Let xi = pqii (qi is coprime to pi ), then pi must be an odd number because negative number has no real square root. From x1 x2 · · · xn = 1, we have p1 p2 · · · pn = q1 q2 · · · qn , so qi is odd number and (−1)xi = −1 (i = 1, 2, · · · , n). In this case, the equation (4) become the following equation: 1 x1 1 1 |a| + |a|x2 + · · · + |a|xn = n|a|. x1 x2 xn From the conclusion of case 1) we know that the theorem is also holds. This completes the proof of the theorem.

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Chuan Lv

No. 2

References [1] F. Smarandache, Only problems,not solutions, Xiquan Publishing House, Chicago, 1993, pp. 22.

[2] Zhang Wenpeng, On an equation of Smarandache and its integer solutions, Smarandache Notions (Book series), American Research Press, 13(2002), 176-178.

[3] Tom M.Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.

[4] R.K.Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1981.

[5] â&#x20AC;&#x153;Smarandache Diophantinc Equationsâ&#x20AC;?at http://www.gallup.unm.edu/ smarandache/ Dioph-Eq.txt.