论文信息 - On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$f(z)=f(x)f(y)=f(u)f(v)

On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$f(z)=f(x)f(y)=f(u)f(v)

We show that the Diophantine system $$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$f(z)=f(x)f(y)=f(u)f(v)has infinitely many nontrivial positive integer solutions for $$f(X)=X^2-1$$f(X)=X2-1, and infinitely many nontrivial rational solutions for $$f(X)=X^2+b$$f(X)=X2+b with nonzero integer b.

Yong Zhang | Zhongyan Shen | Yong Zhang | Zhongyan Shen

[1] Yong Zhang,et al. On products of consecutive arithmetic progressions , 2015 .

[2] Y. Bugeaud. On the diophantine equation (xk − 1)(yk − 1) = (zk − 1) , 2004 .

[3] Deyi Chen,et al. The decomposition of triangular numbers , 2013 .

[4] M. Ulas,et al. On products of disjoint blocks of arithmetic progressions and related equations , 2016, 1601.05032.

[5] M. Bennett. The Diophantine equation (xk − 1)(yk − 1) = (zk − 1)t , 2007 .

[6] Yong Zhang,et al. A note on the Diophantine equation $$f(x)f(y)=f(z^2)$$f(x)f(y)=f(z2) , 2015, Period. Math. Hung..