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Question Number 151322 by mathdanisur last updated on 20/Aug/21

$$\mathrm{Determine}\mathrm{all}\mathrm{pairs}(\mathrm{x};\mathrm{y})\mathrm{of}\mathrm{integers}$$$$\mathrm{which}\mathrm{satisfy}:$$$${\mathrm{x}}^3-3\mathrm{x}+\mathrm{y}=0$$

Commented by mathdanisur last updated on 20/Aug/21

$$\mathrm{with}\mathbf{x}\mathrm{real},\mathbf{y}\mathrm{integer}$$

Answered by Rasheed.Sindhi last updated on 21/Aug/21

$${\mathrm{x}}^3-3\mathrm{x}+\mathrm{y}=0;\mathrm{x}\in \mathbb{R},\mathrm{y}\in \mathbb{Z}$$$$\mathrm{x}({\mathrm{x}}^2-3)=\mathrm{y}$$$$\mathrm{y}=0\Rightarrow \mathrm{x}=0,3when\mathrm{xy}\in \mathbb{Z}$$$$\mathrm{y}=\pm 1\Rightarrow {\mathrm{x}}^3-3\mathrm{x}\mp 1=0$$$$\mathrm{y}=\pm 2\Rightarrow {\mathrm{x}}^3-3\mathrm{x}\mp 2=0$$$$....$$$$...$$$$Infinitesolutions.$$

Commented by mathdanisur last updated on 20/Aug/21

$$\mathrm{Thank}\mathrm{You}\mathbf{S}\mathrm{er}$$

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