Five Fives/Table of 5s

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

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

$1={\dfrac {5}{5}}$

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

$3=\phi ^{5}-5\phi$

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

$5={\sqrt {5\times 5}}$

$10=5+5$

$11=\phi ^{5}-\phi ^{-5}$

$24={\dfrac {5!}{5}}$

$25=5\times 5$

$55$

$115=5!-5$

$125=5!+5$

$600=5\times 5!$

$3125=5^{5}$

$11981655542024930675232002=\phi ^{5!}-\phi ^{-5!}$

$12696403353658275925965100847566516959580321051449436762275840000000000000=55!$

Three 5s

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

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

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

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

$3=\log _{5}{(5!+5)}$

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

$5=5-5+5$

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

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

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

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

$8=\phi ^{5}-5\phi +5$

$9=5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}$

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

$11={\dfrac {55}{5}}$

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

$15=5+5+5$

$16=\phi ^{5}-\phi ^{-5}+5$

$19={\dfrac {5!}{5}}-5$

$20=5\times 5-5$

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

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

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

$25={\sqrt {5}}\left(\phi ^{5}+{\dfrac {1}{\phi ^{5}}}\right)$

$30=5\times 5+5$

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

$55=5\left(\phi ^{5}-{\dfrac {1}{\phi ^{5}}}\right)$

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

$95=5!-5\times 5$

$110=5!-5-5$

$120=5!+5-5$

$123=\phi ^{5}-5\phi +5!$

$130=5!+5+5$

$145=5!+5\times 5$

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

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

$3005=5^{5}-5!$

$503284375=55^{5}$

Four 5s

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

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

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

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

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

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

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

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

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

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

$15=\left(\log _{5}{(5!+5)^{5}}\right)$

$20=5+5+5+5$

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

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

$35=5\times 5+5+5$

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

$50=5\times 5+5\times 5$

$52=55-\phi ^{5}+5\phi$

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

$70=5!+5-55$

$120=5\times 5\times 5-5$

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

$130=5\times 5\times 5+5$

$243=\left(\log _{5}{(5!+5)}\right)^{5}$

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

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

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

$605=55\left(\phi ^{5}-\phi ^{-5}\right)$

$3000=\left(5^{5}-5\right)-5!$

$3025=55\times 55$

$3070=5^{5}-55$

$3100=5^{5}-5\times 5$

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

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