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Updated link to accepted answer

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edited Jan 22, 2018 at 17:02

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For everybody who wants to laugh at my unnecessarily complicated answer (Beware! This will spoiler the right answer to this follow-up question.):

$2008 = 2207 - 199$,
where $2207 = L(16)$ and $199 = L(11)$ – the $16$th and $11$th Lucas numbers, respectively. The $n$th Lucas number can be expressed by $\Phi$ as $$L(n) = \Phi^n + \left( - \Phi \right)^{-n}.$$ With this $$\begin{align} 2008 &= \overbrace{\left( \Phi^{16} + \Phi^{-16} \right)}^{2207} - \overbrace{\left( \Phi^{11} - \Phi^{-11} \right)}^{199} = \Phi^{10} \left( \Phi^6 - \Phi \right) + \Phi^{-12} \left( \Phi + \frac{\Phi}{\Phi^5} \right) \\ &= \Phi^{\frac{\Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi + \Phi + \Phi}{\Phi}} \left( \Phi \Phi \Phi \Phi \Phi \Phi - \Phi \right) + \Phi^{- \frac{\Phi + \Phi + \Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi}{\Phi}} \left( \Phi + \frac{\Phi}{\Phi \Phi \Phi \Phi \Phi} \right). \end{align}$$ Unfortunately this takes 34 $\Phi$s – much more than the accepted answer accepted answer.

For everybody who wants to laugh at my unnecessarily complicated answer (Beware! This will spoiler the right answer to this follow-up question.):

$2008 = 2207 - 199$,
where $2207 = L(16)$ and $199 = L(11)$ – the $16$th and $11$th Lucas numbers, respectively. The $n$th Lucas number can be expressed by $\Phi$ as $$L(n) = \Phi^n + \left( - \Phi \right)^{-n}.$$ With this $$\begin{align} 2008 &= \overbrace{\left( \Phi^{16} + \Phi^{-16} \right)}^{2207} - \overbrace{\left( \Phi^{11} - \Phi^{-11} \right)}^{199} = \Phi^{10} \left( \Phi^6 - \Phi \right) + \Phi^{-12} \left( \Phi + \frac{\Phi}{\Phi^5} \right) \\ &= \Phi^{\frac{\Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi + \Phi + \Phi}{\Phi}} \left( \Phi \Phi \Phi \Phi \Phi \Phi - \Phi \right) + \Phi^{- \frac{\Phi + \Phi + \Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi}{\Phi}} \left( \Phi + \frac{\Phi}{\Phi \Phi \Phi \Phi \Phi} \right). \end{align}$$ Unfortunately this takes 34 $\Phi$s – much more than the accepted answer.

For everybody who wants to laugh at my unnecessarily complicated answer (Beware! This will spoiler the right answer to this follow-up question.):

$2008 = 2207 - 199$,
where $2207 = L(16)$ and $199 = L(11)$ – the $16$th and $11$th Lucas numbers, respectively. The $n$th Lucas number can be expressed by $\Phi$ as $$L(n) = \Phi^n + \left( - \Phi \right)^{-n}.$$ With this $$\begin{align} 2008 &= \overbrace{\left( \Phi^{16} + \Phi^{-16} \right)}^{2207} - \overbrace{\left( \Phi^{11} - \Phi^{-11} \right)}^{199} = \Phi^{10} \left( \Phi^6 - \Phi \right) + \Phi^{-12} \left( \Phi + \frac{\Phi}{\Phi^5} \right) \\ &= \Phi^{\frac{\Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi + \Phi + \Phi}{\Phi}} \left( \Phi \Phi \Phi \Phi \Phi \Phi - \Phi \right) + \Phi^{- \frac{\Phi + \Phi + \Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi}{\Phi}} \left( \Phi + \frac{\Phi}{\Phi \Phi \Phi \Phi \Phi} \right). \end{align}$$ Unfortunately this takes 34 $\Phi$s – much more than the accepted answer.

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created Jan 22, 2018 at 0:20

A. P.

5.9k
1
22
52

For everybody who wants to laugh at my unnecessarily complicated answer (Beware! This will spoiler the right answer to this follow-up question.):

$2008 = 2207 - 199$,
where $2207 = L(16)$ and $199 = L(11)$ – the $16$th and $11$th Lucas numbers, respectively. The $n$th Lucas number can be expressed by $\Phi$ as $$L(n) = \Phi^n + \left( - \Phi \right)^{-n}.$$ With this $$\begin{align} 2008 &= \overbrace{\left( \Phi^{16} + \Phi^{-16} \right)}^{2207} - \overbrace{\left( \Phi^{11} - \Phi^{-11} \right)}^{199} = \Phi^{10} \left( \Phi^6 - \Phi \right) + \Phi^{-12} \left( \Phi + \frac{\Phi}{\Phi^5} \right) \\ &= \Phi^{\frac{\Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi + \Phi + \Phi}{\Phi}} \left( \Phi \Phi \Phi \Phi \Phi \Phi - \Phi \right) + \Phi^{- \frac{\Phi + \Phi + \Phi + \Phi}{\Phi} \cdot \frac{\Phi + \Phi + \Phi}{\Phi}} \left( \Phi + \frac{\Phi}{\Phi \Phi \Phi \Phi \Phi} \right). \end{align}$$ Unfortunately this takes 34 $\Phi$s – much more than the accepted answer.