Tetrational numbers

This page contains numbers of the form ³n that used to have articles on Googology Wiki.
== List of tetrational numbers ==
³4 or to-the-third-four is a positive integer equal to \(4^{4^4}\) or \(4^{256}\), four-to-the-two-hundred-and-fifty-six, with 155 digits. Its full decimal expansion is
The superscript 3 represents tetration. For comparison, the limit of a double is \(2^{1024}-2^{971}\), which is about the square of this number.
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³5 or to-the-third-five is a positive integer equal to \(5^{5^5}\) or \(5^{3,125}\). It has 2,185 digits; its full decimal expansion is
The superscript 3 represents tetration. It is an apocalypenta number.
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³6 (read as "to-the-third-six") is a positive integer equal to \(6^{6^6} = 6^{46,656} \approx 2.659119772153226779682489 \times 10^{36,305}\). It has 36,306 digits; its full decimal expansion is:
The superscript 3 represents tetration.
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³7 (read as "to-the-third-seven") is a positive integer equal to \(7^{7^7} = 7^{823,543} \approx 3.759823526783788538922131 \times 10^{695,974}\). It has 695,975 digits. The superscript 3 represents tetration, meaning this is 7 tetrated to 3.
The full decimal expansion of ³7 is available on this page. (warning: may freeze or crash browsers, particularly for low-end devices)
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³8 (read as "to-the-third-eight") is a positive integer equal to \(8^{8^{8}} = 8^{16,777,216} \approx 6.014520753651392 \times 10^{15,151,335}\). It has 15,151,336 digits. The superscript 3 represents tetration, meaning this is 8 tetrated to 3.
The full decimal expansion of ³8 is available on this page. (warning: you may need a pdf reader application and it can also crash your reader)
== Approximations ==
For 4^4^4:
Notation 	Lower bound 	Upper bound
Scientific notation 	\(1.340\times10^{154}\) 	\(1.341\times10^{154}\)
Arrow notation 	\(4\uparrow\uparrow3\), \(2\uparrow512\) 	\(4\uparrow\uparrow3\), \(2\uparrow512\)
Down-arrow notation 	\(4\downarrow\downarrow5\) 	\(4\downarrow\downarrow5\)
Steinhaus-Moser Notation 	80[3] 	81[3]
Copy notation 	1[155] 	2[155]
Chained arrow notation 	\(4\rightarrow3\rightarrow2\), \(2\rightarrow512\)	\(4\rightarrow3\rightarrow2\), \(2\rightarrow512\)
Taro's multivariable Ackermann function	A(3,509) 	A(3,510)
Pound-Star Notation 	#*((676))*8 	#*((677))*8
PlantStar's Debut Notation 	[91] 	[92]
BEAF 	{4,3,2}, {2,512} 	{4,3,2}, {2,512}
Hyper-E notation 	E[4]4#2 	E[4]4#2
Bashicu matrix system 	(0)(0)(0)(0)(0)[65536] 	(0)(0)(0)(0)(0)[65536]
Bird's array notation 	{4,3,2}, {2,512} 	{4,3,2}, {2,512}
Hyperfactorial array notation 	97! 	98!
Strong array notation 	{4,3,2}, {2,512} 	{4,3,2}, {2,512}
Fast-growing hierarchy 	\(f_2(503)\) 	\(f_2(504)\)
Hardy hierarchy 	\(H_{\omega^2}(499)\) 	\(H_{\omega^2}(500)\)
Slow-growing hierarchy 	\(g_{\omega^{\omega^{\omega}}}(4)\) 	\(g_{\omega^{\omega^{\omega}}}(4)\)
== See also ==
* 65,536
* 7,625,597,484,987
* ⁵2
* ³9
* ³10
Category:Numbers
Category:Lists
Category:Tetration
Category:Class 2
Category:Class 3
Category:Numbers with 101 to 999 digits
Category:Numbers with 1000 to 1000000 digits
Category:Lower class 3