Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 295–301

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

• Yong Zhang
• Zhongyan Shen
Article

## Abstract

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

## Keywords

Diophantine system Positive integer solution Rational solution Elliptic curve

## Mathematics Subject Classification

Primary 11D25 Secondary 11D72 11G05

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