Let $$f\in \mathbb {Q}[x]$$ f ∈ Q [ x ] be a polynomial without multiple roots and $$\deg {f}\ge 2$$ deg f ≥ 2 . We give conditions for $$f=x^2+bx+c$$ f = x 2 + b x + c under which the Diophantine equation $$2f(x)f(y)=f(z)(f(x)+f(y))$$ 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 $$f=x^2+bx$$ f = x 2 + b x with nonzero integer b . Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.

中文翻译：

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具有调和平均数的丢番图方程

令 $$f\in \mathbb {Q}[x]$$ f ∈ Q [ x ] 是一个没有多个根的多项式且 $$\deg {f}\ge 2$$ deg f ≥ 2 。我们给出 $$f=x^2+bx+c$$ f = x 2 + bx + c 的条件，其中丢番图方程 $$2f(x)f(y)=f(z)(f(x) +f(y))$$ 2 f ( x ) f ( y ) = f ( z ) ( f ( x ) + f ( y ) ) 有无穷多个非平凡整数解并证明这个方程有无穷多个有理参数解对于 $$f=x^2+bx$$ f = x 2 + bx 与非零整数 b 。此外，我们证明了它对于无穷多个三次多项式具有有理参数解。