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The asankhyeya (also called asaṃkhyeya[1]) is a number described in Buddhist texts that is equal to \(10^{140}\), or 1 followed by 140 zeroes.[2] It is pronounced Asougi in Japanese where it is equal to \(10^{56}\), and means "innumerable".

The Avatamsaka Sutra [1] gives an alternate description of Asankhyeya as \(10^{7\times2^{103}}\), defining a series of numbers that are squares of each other starting with one koti equalling \(10^7\), one koti kotis making an ayuta (\(10^{14}\)), one ayuta ayutas making a nayuta (\(10^{28}\)), and so on, with Asankhyeya being the 104th number in this chain.

Approximations

For 10140:

Notation Lower bound Upper bound
Scientific notation \(1\times10^{140}\)
Arrow notation \(10\uparrow140\)
Steinhaus-Moser Notation 74[3] 75[3]
Copy notation 9[140] 1[141]
Taro's multivariable Ackermann function A(3,462) A(3,463)
Pound-Star Notation #*(1,2,8,11,9,8,5)*12 #*(4,4,10,5,7,2,5,2)*10
BEAF {10,140}
Hyper-E notation E140
Bashicu matrix system (0)(0)(0)(0)(0)[23713] (0)(0)(0)(0)(0)[23714]
Hyperfactorial array notation 90! 91!
Fast-growing hierarchy \(f_2(456)\) \(f_2(457)\)
Hardy hierarchy \(H_{\omega^2}(456)\) \(H_{\omega^2}(457)\)
Slow-growing hierarchy \(g_{\omega^{\omega^2+\omega4}}(10)\)

For 107×2103:

Notation Lower bound Upper bound
Arrow notation \((10\uparrow7)\uparrow2\uparrow103\)
Down-arrow notation \(57\downarrow\downarrow19\) \(715\downarrow\downarrow12\)
Steinhaus-Moser Notation 22[3][3] 23[3][3]
Copy notation 6[6[32]] 7[7[32]]
H* function H(23H(9)) H(24H(9))
Taro's multivariable Ackermann function A(3,A(3,104)) A(3,A(3,105))
Pound-Star Notation #*((1))*(1,10,10)*4 #*((1))*(5,2,1)*6
BEAF {{10,7},{2,103}}
Hyper-E notation E(7E[2]103)
Bashicu matrix system (0)(1)[10] (0)(1)[11]
Hyperfactorial array notation (28!)! (29!)!
Fast-growing hierarchy \(f_2(f_2(100))\) \(f_2(f_2(101))\)
Hardy hierarchy \(H_{\omega^22}(100)\) \(H_{\omega^22}(101)\)
Slow-growing hierarchy \(g_{\omega^{\omega^{\omega3+1}7}}(10)\) \(g_{\omega^{\omega^{\omega3+1}8}}(10)\)

Sources

  1. 1.0 1.1 "How large is one Asamkhyeya?" Bodhi Field. http://www.drbachinese.org/vbs/publish/462/vbs462p042.pdf
  2. [1]

See also

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