From charlesreid1

Back to Five Fives

One 5

Various ways of arranging a single 5 to yield different numbers. (More limited than 4, of course...)


5^{\frac{1}{2}} = \sqrt{5}


5 = 5


120 = 5!


Two 5s


0 = \ln{ \dfrac{5}{5} }


1 = \dfrac{5}{5}


2 = \dfrac{ \ln{(5)} }{ \ln{(\sqrt{5})}  }


4 = \dfrac{ \ln{5} }{ \ln{\left( \sqrt{ \sqrt{ 5 } } \right) } }


5 = \sqrt{5 \times 5}


10 = 5 + 5


11 = \phi^5 - \dfrac{1}{\phi^5}


24 = \dfrac{5!}{5}


25 = 5 \times 5


55


115 = 5! - 5


125 = 5! + 5


600 = 5 \times 5!


3125 = 5^5


12696403353658275925965100847566516959580321051449436762275840000000000000 = 55!

Three 5s


\dfrac{1}{2} = \dfrac{      \ln{ \left( \dfrac{5}{\sqrt{5}} \right) }     }{    \ln{(5)}     }


\dfrac{25}{24} = \dfrac{5! + 5}{5!}


1 = \dfrac{ \sqrt{5 \times 5} }{ 5 }


2 = \dfrac{    \ln{(5)}     }{      \ln{ \left( \dfrac{5}{\sqrt{5}} \right) }     }


3 = \log_{5}{(5! + 5)}


4 = 5 - \dfrac{5}{5}


5 = 5 - 5 + 5


5 = \sqrt[5]{5^5}


5 = \sqrt{ \dfrac{ \sqrt{5^5} }{ \sqrt{5} } }


5 = \dfrac{5 \times 5}{5}


6 = 5 + \dfrac{5}{5}


10 = \sqrt{5} \left( \sqrt{5} + \sqrt{5} \right)


11 = \dfrac{55}{5}


12 = \dfrac{5!}{5+5}


15 = 5 + 5 + 5


16 = \phi^{5} - \dfrac{1}{\phi^5} + 5


20 = 5 \times 5 - 5


20 = \dfrac{ 5 \lg{5} }{ \lg{\left(\sqrt{\sqrt{5}}\right)} }


30 = 5 \times 5 + 5


32 = \left( \dfrac{ \ln{5} }{ \ln{ \sqrt{ 5 } } } \right)^5


25 = \dfrac{ \sqrt{5^5} }{ \sqrt{5} }


60 = \dfrac{5!}{ \left( \dfrac{\ln{5}}{\ln{\sqrt{5}}} \right) }


95 = 5! - 5 \times 5


130 = 5! + 5 + 5


145 = 5! + 5 \times 5


625 = \dfrac{5^5}{5}


1024 = \left( \dfrac{ \ln{5} }{ \ln{ \sqrt{ \sqrt{5} } } } \right)^5


503284375 = 55^5

Four 5s


1 = \dfrac{  \ln{ \left( \dfrac{5}{ \sqrt{5} } \right) }  + \ln{5}   }{   \ln{5}   }


1 = \dfrac{5^5}{5^5}


1 = \dfrac{ 5 \times 5}{5 \times 5}


6 = \dfrac{ \ln{(5)} }{ \ln{(\sqrt{5})} } + \dfrac{ \ln{(5)} }{ \ln{(\sqrt{\sqrt{5}})} }


7 = \sqrt{ \dfrac{5!}{5} + (5 \times 5) }


8 = \left( \dfrac{ \ln{(5)} }{ \ln{(\sqrt{5})} } \right) \left( \dfrac{ \ln{(5)} }{ \ln{(\sqrt{\sqrt{5}})} } \right)


9 = 5 + 5 - \dfrac{5}{5}


11 = 5 + 5 + \dfrac{5}{5}


12 = \left( \dfrac{5!}{5} \right) \times \left( \dfrac{ \ln{\sqrt{5}} }{ \ln{5} } \right)


13 = \dfrac{5! - 55}{5}


15 = \left( \log_{5}{ (5! + 5)^5 } \right)


20 = 5 + 5 + 5 + 5


26 = \dfrac{5^5 - 5}{5!}


35 = \dfrac{55 + 5!}{5}


35 = 5 \times 5 + 5 + 5


49 = \dfrac{5!}{5} + (5 \times 5)


50 = 5 \times 5 + 5 \times 5


60 = 5 \left( \dfrac{ 5! }{ 5 + 5 } \right)


70 = 5! + 5 - 55


120 = 5 \times 5 \times 5 - 5


123 = \phi^{5+5} + \dfrac{1}{\phi^{5+5}}


130 = 5 \times 5 \times 5 + 5


243 = \left( \log_{5}{(5! + 5)} \right) ^5


( 5_{5} )! = (10_{5})! = 440_{5} = 120_{10}

(5 base 5 is 10 base 5; 5 base 5 factorial is 440 base 5, or 120 base 10. No surprises here.)


505 = \dfrac{5^5}{5} - 5!


3025 = 55 \times 55


3125 = \dfrac{5 \times 5^5}{5}


161051 = 11^5 = \left( \dfrac{55}{5} \right)^5