Question and Answers Forum

All Questions      Topic List

Logic Questions

Previous in All Question      Next in All Question      

Previous in Logic      Next in Logic      

Question Number 134689 by benjo_mathlover last updated on 06/Mar/21

$$\frac{2^{289}+1}{2^{17}+1}=2^{a_1}+2^{a_2}+2^{a_3}+...+2^{a_k}$$$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}k.$$

Answered by EDWIN88 last updated on 06/Mar/21

$$\mathrm{Let}{2}^{17}=\mathrm{y}$$$$\Rightarrow \frac{{\mathrm{y}}^{17}+1}{\mathrm{y}+1}={\mathrm{y}}^{16}-{\mathrm{y}}^{15}+{\mathrm{y}}^{14}-{\mathrm{y}}^{13}+{\mathrm{y}}^{12}-{\mathrm{y}}^{11}+...+1$$$$={\mathrm{y}}^{15}(\mathrm{y}-1)+{\mathrm{y}}^{13}(\mathrm{y}-1)+{\mathrm{y}}^{11}(\mathrm{y}-1)+...+\mathrm{y}(\mathrm{y}-1)+1$$$$=({\mathrm{y}}^{15}+{\mathrm{y}}^{13}+{\mathrm{y}}^{11}+{\mathrm{y}}^9+...+\mathrm{y})(\mathrm{y}-1)+1$$$$=(2^{255}+2^{221}+2^{187}+...+2^{17})(2^{17}-1)+1$$$$=\underset{8}{\underbrace{(2^{255}+2^{221}+2^{187}+...+2^{17})}\underset{17}{(2^{16}+2^{15}+2^{14}+...+2+1)}}+1$$$$\mathrm{so}\mathrm{we}\mathrm{get}k=8\times 17+1=137$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com