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Question Number 76952 by ~blr237~ last updated on 01/Jan/20
show that   (√(1+ 2017×2018×2019×2020 )) ∈ N
$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\sqrt{\mathrm{1}+\:\mathrm{2017}×\mathrm{2018}×\mathrm{2019}×\mathrm{2020}\:}\:\in\:\mathbb{N} \\ $$
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   ⇒ (√(1+x(x+1)(x+2)(x+3)))=∣x^2 +3x+1∣  ∣x^2 +3x+1∣∈N∀x∈Z  in our case x=2017∈Z ⇒ (√(1+2017×2018×2019×2020))∈N
$$\mathrm{1}+{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)= \\ $$$$={x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{3}} +\mathrm{11}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{1}= \\ $$$$=\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\:\sqrt{\mathrm{1}+{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)}=\mid{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\mid \\ $$$$\mid{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{1}\mid\in\mathbb{N}\forall{x}\in\mathbb{Z} \\ $$$$\mathrm{in}\:\mathrm{our}\:\mathrm{case}\:{x}=\mathrm{2017}\in\mathbb{Z}\:\Rightarrow\:\sqrt{\mathrm{1}+\mathrm{2017}×\mathrm{2018}×\mathrm{2019}×\mathrm{2020}}\in\mathbb{N} \\ $$

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