Super-leviathan is \(9^{9^{9}}\), which is the largest number that can be written using only three digits.[1] By using tetration, it can be written as 39 (read as "to-the-third-nine"). It is equal to \(9^{387,420,489} \approx 4.281247731757470 \times 10^{369,693,099}\). It has 369,693,100 digits.
Cookiefonster calls this number Ulysses Number.[2]
This number is described in the novel Ulysses by James Joyce, who wrote:
Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e. g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers.[3][4]
Full decimal expansion is calculated.[5][6][7]
Approximations[]
| Notation | Lower bound | Upper bound |
|---|---|---|
| Scientific notation | \(4.281\times10^{369\,693\,099}\) | \(4.282\times10^{369\,693\,099}\) |
| Arrow notation | \(9\uparrow\uparrow3\) | |
| Steinhaus-Moser Notation | 8[3][3] | 9[3][3] |
| Copy notation | 3[3[9]] | 4[4[9]] |
| Taro's multivariable Ackermann function | A(3,A(3,27)) | A(3,A(3,28)) |
| Pound-Star Notation | #*((1))*6798 | #*((1))*6799 |
| BEAF | {9,3,2} | |
| Hyper-E notation | E[9]1#3 | |
| Bashicu matrix system | (0)(1)[5] | (0)(1)[6] |
| Hyperfactorial array notation | (11!)! | (12!)! |
| Fast-growing hierarchy | \(f_2(f_2(25))\) | \(f_2(f_2(26))\) |
| Hardy hierarchy | \(H_{\omega^22}(25)\) | \(H_{\omega^22}(26)\) |
| Slow-growing hierarchy | \(g_{\omega^{\omega^\omega}}(9)\) | |
Sources[]
- ↑ Clifford Pickover. Keys to Infinity Wiley. 1995. ISBN 978-0471118572 p.102
- ↑ Cookiefonster. Pointless Gigantic List of Numbers - Part 2 (1,000,000 ~ 10^10^1,000,000)
- ↑ https://en.wikisource.org/wiki/Page:Ulysses,_1922.djvu/655
- ↑ https://web.archive.org/web/20240511223812/http://mrob.com/pub/math/numbers-21.html
- ↑ Hans Havermann. 9^9^9 in 33 volumes 2004-01-06.
- ↑ Fish. Full decimal expansion of super-leviathan 2024-09-23.
- ↑ OEIS. A241298 Decimal expansion of 9^(9^9) = 9^^3.
See also[]
Specific numbers: Beast number · Belphegor's prime · Legion's number of the first kind · Super-leviathan · Legion's number of the second kind · Leviathan number
Forms of numbers: apocalypse number · apocalyptic number · apocalyptri number · apocalyptetra number · apocalypenta number · apocalyhexakosioihexekontahexa number · Goliath number