This page contains unnamed powers of 2 that used to have articles on the Googology Wiki. The former content of these articles is also included here.

List of powers of 2

512 is equal to 29. 512 is a Dudeney number, that is it is equal to the cube of the sum of its digits. It is the smallest base-10 Dudeney number (the others are 4,913, 5,832, 17,576, and 19,683).

In the Bignum Bakeoff contest, entries are limited to 512 characters (ignoring whitespace)


210 = 1,024


A screenshot from the game 2048

2,048 is a positive integer following 2,047 and preceding 2,049. It is the 11th power of 2, and equal to f2(8) and f3(2) in the fast-growing hierarchy.

It is also the largest known power of 2 where all digits are even.[1]

2048 is also the name of a web game where you slide tiles representing powers of 2 across a board and try to combine them to reach higher and higher powers of 2. The largest tile you can reach is 131072.


The 131072 tile, screenshotted from this video.

131,072 is the 17th power of 2.

In the 2048 game, it is the largest tile you can reach.


262,144 is the fourth expofactorial number, equal to 4321 = 432 = 49.


536,870,912 is the largest power of two number with distinct digits.[2] It is just over half a billion.


2,147,483,648 is a positive integer equal to \(2^{31} = 2^{2^5 - 1}\). It is notable in computer science for being the absolute value of the maximum negative value of a 32-bit signed integer, which have the range [-2147483648, 2147483647].

Its full name in English is "two billion/milliard one hundred forty-seven million four hundred eighty-three thousand six hundred forty-eight," where the short scale uses "billion" and the long scale uses "milliard."


286 = 77,371,252,455,336,267,181,195,264 is the largest known power of two not containing a zero.


2168 = 374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856 is the largest known power of two not containing all decimal digits (in this case, the digit 2 is missing).[3]


2219 = 842,498,333,348,457,493,583,344,221,469,363,458,551,160,763,204,392,890,034,487,820,288 is the largest power of two with an accepted English name.


21,000,000 ≈ 9.90065623 × 10301,029

Approximations of these numbers

For 131,072:

Notation Lower bound Upper bound
Scientific notation \(1.31072 \times 10^5\) (exact)
Arrow notation \(2↑17\) (exact)
Steinhaus-Moser Notation 6[3] 7[3]
Copy notation 1[6] 2[6]
Taro's multivariable Ackermann function A(3,14) A(3,15)
Pound-Star Notation #*(36)*3 #*(37)*3
BEAF \(\{2,17\}\)
Hyper-E notation E[2]17
Hyperfactorial array notation 8! 4!1
Fast-growing hierarchy \(f_2(13)\) \(f_2(14)\)
Hardy hierarchy \(H_{\omega^2}(13)\) \(H_{\omega^2}(14)\)
Slow-growing hierarchy \(g_{\omega^{17}}(2)\)

For 262,144:

Notation Lower bound Upper bound
Scientific notation \(2.62144 \times 10^5\)
Arrow notation \(2\uparrow18\), \(4\uparrow9\)
Steinhaus-Moser Notation 6[3] 7[3]
Copy notation 2[6] 3[6]
Chained arrow notation \(2\rightarrow18\), \(4\rightarrow9\)
Taro's multivariable Ackermann function A(3,14) A(3,15)
Pound-Star Notation #*(16)*4
PlantStar's Debut Notation [3] [4]
BEAF \(\{2,18\}, \{4,9\}\)
Hyper-E notation E[2]18, E[4]9
Bashicu matrix system (0)[512]
Hyperfactorial array notation 4!1
Bird's array notation \(\{2,18\}, \{4,9\}\)
Strong array notation s(2,18), s(4,9)
Fast-growing hierarchy \(f_2(14)\) \(f_2(15)\)
Hardy hierarchy \(H_{\omega^2}(14)\) \(H_{\omega^2}(15)\)
Slow-growing hierarchy \(g_{\omega^{\omega 2 +1}}(4)\)

For 536,870,912:

Notation Lower bound Upper bound
Scientific notation \(5.36870912\times10^8\) (exact)
Arrow notation \(2↑29\) (exact)
Steinhaus-Moser Notation 9[3] 10[3]
Copy notation 4[9] 5[9]
Taro's multivariable Ackermann function A(3,25) A(3,26)
Pound-Star Notation #*(99)*4 #*(38)*5
BEAF {2,29}
Hyper-E notation E[2]29
Hyperfactorial array notation 12! 13!
Fast-growing hierarchy \(f_2(24)\) \(f_2(25)\)
Hardy hierarchy \(H_{\omega^2}(24)\) \(H_{\omega^2}(25)\)
Slow-growing hierarchy \(g_{\omega^{29}}(2)\)

For 2,147,483,648:

Notation Lower bound Upper bound
Scientific notation \(2.147483648\times10^9\) (exact)
Arrow notation \(2 \uparrow 31\) (exact)
Steinhaus-Moser Notation 9[3] 10[3]
Copy notation 21[5] 2[10]
Taro's multivariable Ackermann function A(3,28) A(3,29)
Pound-Star Notation #*(2,2,3)*3 #*(24)*6
BEAF {2,31}
Hyperfactorial array notation 12! 13!
Fast-growing hierarchy \(f_2(26)\) \(f_2(27)\)
Hardy hierarchy \(H_{\omega^2}(26)\) \(H_{\omega^2}(27)\)
Slow-growing hierarchy \(g_{\omega^{31}}(2)\)

Sources

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