Menu Close

Find-mininum-value-of-n-such-that-both-n-3-and-2020n-1-are-square-numbers-




Question Number 84553 by naka3546 last updated on 14/Mar/20
Find  mininum  value  of  n  such  that  both  n + 3   and  2020n + 1  are  square  numbers .
$${Find}\:\:{mininum}\:\:{value}\:\:{of}\:\:{n}\:\:{such}\:\:{that} \\ $$$${both}\:\:{n}\:+\:\mathrm{3}\:\:\:{and}\:\:\mathrm{2020}{n}\:+\:\mathrm{1}\:\:{are}\:\:{square}\:\:{numbers}\:. \\ $$
Commented by mr W last updated on 14/Mar/20
i got n_(min) =2022  2022+3=2025=45^2  ⇒ok  2020×2022+1=(2021−1)(2021+1)+1=2021^2  ⇒ok
$${i}\:{got}\:{n}_{{min}} =\mathrm{2022} \\ $$$$\mathrm{2022}+\mathrm{3}=\mathrm{2025}=\mathrm{45}^{\mathrm{2}} \:\Rightarrow{ok} \\ $$$$\mathrm{2020}×\mathrm{2022}+\mathrm{1}=\left(\mathrm{2021}−\mathrm{1}\right)\left(\mathrm{2021}+\mathrm{1}\right)+\mathrm{1}=\mathrm{2021}^{\mathrm{2}} \:\Rightarrow{ok} \\ $$
Commented by naka3546 last updated on 14/Mar/20
how  about  n = 726 ?
$${how}\:\:{about}\:\:{n}\:=\:\mathrm{726}\:? \\ $$
Commented by mr W last updated on 14/Mar/20
yes. 726 is the smallest n.
$${yes}.\:\mathrm{726}\:{is}\:{the}\:{smallest}\:{n}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *