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65,536 is a positive integer equal to $$2^{16} = 2^{2^{2^{2}}} = 2 \uparrow\uparrow 4 = 2 \uparrow\uparrow\uparrow 3$$. It is the largest power tower of 2's that can be expressed in a compact digital form (as $$2 \uparrow\uparrow 5$$ is an integer of 19,729 decimal digits). It is also the only nontrivial result of pentation expressible in compact digital form, and 1 of the 2 nontrivial results of pentation expressible in decimal (the other is $$3\uparrow^32$$) — $$^52$$ has 19,729 decimal digits, and $$^62$$ has well over a googol decimal digits.

It is one less than the largest known Fermat prime, 65537.

It is the largest known power of two which does not contain the digits 1, 2, 4, or 8.

It is found commonly in computer science, being the number of different values expressible in 16 bits (a short integer).

Its full name in English is "sixty-five thousand five hundred thirty-six."

Sbiis Saibian calls this number octal-clover mite.[1]

Username5243 calls this number Binary-Dooquol or Binary-Tootol, and it's equal to 2[2]4 = 2[3]3 in Username5243's Array Notation.[2]

Licorneuhh also calls this number Tails-Bi-3-primol, and it's equal to Я***(2) using Primarial Functions.[3]

## In googological notations

There are many notations where 65,536 can be expressed:

Notation Lower bound Upper bound
Scientific notation $$6.5536\times10^4$$
Arrow notation $$2\uparrow\uparrow4$$
Steinhaus-Moser Notation 6[3] 7[3]
Copy notation 5[5] 6[5]
Taro's multivariable Ackermann function A(3,13) A(3,14)
Pound-Star Notation #*(27)*3 #*(28)*3
BEAF {2,4,2}
Hyper-E notation E[2]1#4
Bashicu matrix system (0)[256]
Hyperfactorial array notation 8! 4!1
Fast-growing hierarchy $$f_2(12)$$ $$f_2(13)$$
Hardy hierarchy $$H_{\omega^2}(12)$$ $$H_{\omega^2}(13)$$
Slow-growing hierarchy $$g_{\omega^{\omega2}}(4)$$

Wiki User Plain'N'Simple showed that 65536 can be expressed in the language of FOST in at most 1219 symbols. After building on the work of several other users, including 12AbBa and P bot, this bound was reduced to 991 symbols. Therefore, Rayo(991) is at least 65537.

## Sources

1. Saibian, Sbiis. Hyper-E Numbers. Retrieved 2016-07-17.
2. Part 1 - My Large Numbers
3. Primarial functions numbers | Licorneuhh's numbers site (retrieved at UTC 16:20 12/02/2021)

-minutia group
n-ary numbers with multipliers
Eyelash mite group: eyelash mite-speck · eyelash mite-crumb · eyelash mite-chunk · (binary/ternary/octal)eyelash mite · eyelash mite-bunch · eyelash mite-crowd · eyelash mite-swarm
Dust mite group: dust mite-speck · dust mite-crumb · dust mite-chunk · (binary/ternary/octal)dust mite · dust mite-bunch · dust mite-crowd · dust mite-swarm
Cheese mite group: cheese mite-speck · cheese mite-crumb · cheese mite-chunk · (binary/ternary/octal)cheese mite · cheese mite-bunch · cheese mite-crowd · cheese mite-swarm
Clover mite group: clover mite-speck · clover mite-crumb · clover mite-chunk · (binary/ternary/octal)clover mite · clover mite-bunch · clover mite-crowd · clover mite-swarm
Pipsqueak group: (binary/ternary/octal)pipsqueak
Little squeaker group: (binary/ternary/octal)little squeaker
Small fry group: (binary/ternary/octal)small fry
Guppy group: (binary/ternary/octal)guppyspeck · (binary/ternary/octal)guppycrumb · (binary/ternary/octal)guppychunk · (ternary/quaternary/quinary/duodecimal/hexadecimal/vigesimal/sexagesimal)guppy(bit/byte)
Minnow group: (binary/ternary/octal)minnowspeck · (binary/ternary/octal)minnowcrumb · (binary/ternary/octal)minnowchunk · (ternary)minnow(bit/byte)
Goby group: (binary/ternary/octal)gobyspeck · (binary/ternary/octal)gobycrumb · (binary/ternary/octal)gobychunk · (ternary)goby(bit/byte)