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Question Number 76952 by ~blr237~ last updated on 01/Jan/20

$$\mathrm{show}\mathrm{that}$$$$\sqrt{1+2017\times 2018\times 2019\times 2020}\in \mathbb{N}$$

Answered by MJS last updated on 01/Jan/20

$$1+x(x+1)(x+2)(x+3)=$$$$=x^4+6x^3+11x^2+6x+1=$$$$={(x^2+3x+1)}^2$$$$\Rightarrow \sqrt{1+x(x+1)(x+2)(x+3)}=\mid x^2+3x+1\mid$$$$\mid x^2+3x+1\mid \in \mathbb{N}\mathrm{\forall }x\in \mathbb{Z}$$$$\mathrm{in}\mathrm{our}\mathrm{case}x=2017\in \mathbb{Z}\Rightarrow \sqrt{1+2017\times 2018\times 2019\times 2020}\in \mathbb{N}$$

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