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 $\dfrac{1}{2} = \dfrac{ \ln{ \left( \dfrac{5}{\sqrt{5}} \right) } }{ \ln{(5)} }$ $\dfrac{25}{24} = \dfrac{5! + 5}{5!}$ $1 = \dfrac{ \sqrt{5 \times 5} }{ 5 }$ $2 = \dfrac{ \ln{(5)} }{ \ln{ \left( \dfrac{5}{\sqrt{5}} \right) } }$ $3 = \log_{5}{(5! + 5)}$ $4 = 5 - \dfrac{5}{5}$ $5 = 5 - 5 + 5$ $5 = \sqrt{5^5}$ $5 = \sqrt{ \dfrac{ \sqrt{5^5} }{ \sqrt{5} } }$ $5 = \dfrac{5 \times 5}{5}$ $6 = 5 + \dfrac{5}{5}$ $10 = \sqrt{5} \left( \sqrt{5} + \sqrt{5} \right)$ $11 = \dfrac{55}{5}$ $12 = \dfrac{5!}{5+5}$ $15 = 5 + 5 + 5$ $16 = \phi^{5} - \dfrac{1}{\phi^5} + 5$ $20 = 5 \times 5 - 5$ $20 = \dfrac{ 5 \lg{5} }{ \lg{\left(\sqrt{\sqrt{5}}\right)} }$ $30 = 5 \times 5 + 5$ $32 = \left( \dfrac{ \ln{5} }{ \ln{ \sqrt{ 5 } } } \right)^5$ $25 = \dfrac{ \sqrt{5^5} }{ \sqrt{5} }$ $60 = \dfrac{5!}{ \left( \dfrac{\ln{5}}{\ln{\sqrt{5}}} \right) }$ $95 = 5! - 5 \times 5$ $130 = 5! + 5 + 5$ $145 = 5! + 5 \times 5$ $625 = \dfrac{5^5}{5}$ $1024 = \left( \dfrac{ \ln{5} }{ \ln{ \sqrt{ \sqrt{5} } } } \right)^5$ $503284375 = 55^5$

## 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.) $505 = \dfrac{5^5}{5} - 5!$ $3025 = 55 \times 55$ $3125 = \dfrac{5 \times 5^5}{5}$ $161051 = 11^5 = \left( \dfrac{55}{5} \right)^5$