How to compare 3^3^3^3 and to 3↑(3↑↑3). (Ackerman number, arrow notation) Are these two numbers equal??? How to compare 3↑(3↑↑3) with googol and googolplex???
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Your first two numbers are equal. By definition $3\uparrow\uparrow 3=3^{3^3}$ and $3\uparrow n=3^n$, so $$\large3\uparrow(3\uparrow\uparrow 3)=3\uparrow 3^{3^3}=3^{3^{3^3}}\;.$$ Now $3\uparrow\uparrow 3=7,625,597,484,987$, so $$\log_{10}\big(3\uparrow(3\uparrow\uparrow 3)\big)=7,625,597,484,987\log_{10}3>\big(7\times10^{12}\big)\log_{10}3>3\times10^{12}\;,$$ so $$3\large\uparrow(3\uparrow\uparrow 3)>10^{3\times10^{12}}>10^{100}=1\text{ googol}\;.$$ On the other hand, $$\log_{10}\big(3\uparrow(3\uparrow\uparrow 3)\big)=7,625,597,484,987\log_{10}3<10^{13}\;,$$ so $$3\large\uparrow(3\uparrow\uparrow 3)<10^{10^{13}}<10^{10^{100}}=1\text{ googolplex}\;.$$ |
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