2-289-1-2-17-1-2-a-1-2-a-2-2-a-3-2-a-k-Find-the-value-of-k-

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}} + \dots + 2^{a_{k}}

Find the value of k .

Answered by EDWIN88 last updated on 06/Mar/21

Let 2^{17} = y

\Rightarrow \frac{y^{17} + 1}{y + 1} = y^{16} - y^{15} + y^{14} - y^{13} + y^{12} - y^{11} + \dots + 1

= y^{15} (y - 1) + y^{13} (y - 1) + y^{11} (y - 1) + \dots + y (y - 1) + 1

= (y^{15} + y^{13} + y^{11} + y^{9} + \dots + y) (y - 1) + 1

= (2^{255} + 2^{221} + 2^{187} + \dots + 2^{17}) (2^{17} - 1) + 1

= \underset{8}{\underset{⏟}{(2^{255} + 2^{221} + 2^{187} + \dots + 2^{17})} \underset{17}{(2^{16} + 2^{15} + 2^{14} + \dots + 2 + 1)}} + 1

so we get k = 8 \times 17 + 1 = 137

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