A Diophantine equation with the harmonic mean
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A Diophantine equation with the harmonic mean Yong Zhang1,2 · Deyi Chen3
© Akadémiai Kiadó, Budapest, Hungary 2019
Abstract Let f ∈ Q[x] be a polynomial without multiple roots and deg f ≥ 2. We give conditions for f = x 2 + bx + c under which the Diophantine equation 2 f (x) f (y) = f (z)( f (x) + f (y)) has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for f = x 2 + bx with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials. Keywords Diophantine equation · Pell’s equation · Integer solutions · Rational parametric solutions Mathematics Subject Classification Primary 11D72 · 11D25; Secondary 11D41 · 11G05
1 Introduction Let f ∈ Q[x] be a polynomial without multiple roots and deg f ≥ 2. Several authors have investigated the Diophantine equation f (x) f (y) = f (z)2
(1.1)
(we refer to [1,5,6,8,11–13,16–18]). On the one hand, (1.1) means that the values of f (x), f (z) and f (y) are in a geometric progression. On the other hand, (1.1) can be considered as the geometric mean of f (x), f (y) is f (z). In this direction we can study the harmonic mean of f (x), f (y) is f (z), i.e.,
This research was supported by the National Natural Science Foundation of China (Grant No. 11501052) and Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology).
B
Yong Zhang [email protected] Deyi Chen [email protected]
1
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, People’s Republic of China
2
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, People’s Republic of China
3
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
123
Y. Zhang, D. Chen
2 1 f (x)
+
1 f (y)
= f (z),
(1.2)
the arithmetic mean of f (x), f (y) is f (z), i.e., f (x) + f (y) = f (z), 2 and the quadratic mean of f (x), f (y) is f (z), i.e., f (x)2 + f (y)2 = f (z). 2
(1.3)
(1.4)
Now we mainly care about the integer solutions and rational parametric solutions of (1.2) for quadratic and cubic polynomials. (1.2) is equivalent to 2 f (x) f (y) = f (z)( f (x) + f (y)).
(1.5)
It is worth to note that (1.5) has the same degree of the left and right hand. A similar Diophantine equation f (x) f (y) = f (z 2 ) was studied by Zhang and Cai [14,15]. We call a solution (x, y, z) of (1.2) or (1.5) nontrivial if f (x) f (y)( f (x) + f (y)) = 0. Using the theory of Pell’s equations, we have Theorem 1.1 Let f = x 2 + bx + c be a quadratic polynomial without multiple roots, where b, c are integers. Suppose that (x0 , y0 , z 0 ) is an integer solution of (1.5) and y0 = x0 +2z 0 +b. Then (1.5) has infinitely many nontrivial integer solutions. Using the theory of elliptic curves, we get Theorem 1.2 Let f = x 2 + bx with nonzero integer b, then (1.5) has infinitely many rational parametric soluti
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