**Promaxima**, short for "probability maximum", is a physical constant approximately equal to \(10^{10^{245}}\).^{[1]}^{[2]} It is an upper bound on the number of different possible parallel universes taking only into account our observable universe. The term was coined by Sbiis Saibian in 2004, in response to an anonymous contributor, who wrote:

So I was thinking about large number which actually could possibly somehow correspond to actual real physical world phenomena. Like this. So take the entire space of the universe, and imagine that it was filled *solid* with atoms or subatomic particles. The number of those particles would be huge, I wonder what it'd be. But, now, what about all possible permutations of those particles in space. I guess that'd be N![referring to the factorial]where N is the number of particles. Someone figure out the number! It's gotta approach or exceed Googol, or even Googolplex. Yathink?

Saibian solved this problem by packing the observable universe with "strings," the smallest possible physical objects. These are \(10^{-35}\) meters long, and the diameter of the observable universe is \(10^{26}\) meters. Packing the universe, then, takes \(\left(\frac{10^{26}}{10^{-35}}\right)^3 = 10^{183}\) strings. (Note that *our* universe is not packed with strings; therefore this number is an overestimate.) The number of arrangements of these strings is then \(10^{183}!\), approximately \(10^{10^{185}}\) (using Stirling's approximation).

Saibian took this a step further by introducing time. The number calculated above is only the number of the strings in a single Planck time, or about \(10^{−43}\) seconds. Letting the universe run for 500 quadrillion years, the number of possibilities becomes \(10^{10^{245}}\). Promaxima is a good bit larger than googolplex.

## Approximations

Notation | Lower bound | Upper bound |
---|---|---|

Arrow notation | \(10\uparrow10\uparrow245\) | |

Down-arrow notation | \(10\downarrow\downarrow246\) | |

Steinhaus-Moser Notation | 117[3][3] | 118[3][3] |

Copy notation | 9[9[245]] | 1[10[123]] |

H* function | H(33H(80)) | H(34H(80)) |

Taro's multivariable Ackermann function | A(3,A(3,812)) | A(3,A(3,813)) |

Pound-Star Notation | #*((1))*((5))*8 | #*((1))*((6))*8 |

BEAF | {10,{10,245}} | |

Hyper-E notation | E245#2 | |

Bashicu matrix system | (0)(1)[811] | (0)(1)[812] |

Hyperfactorial array notation | (140!)! | (141!)! |

Fast-growing hierarchy | \(f_2(f_2(805))\) | \(f_2(f_2(806))\) |

Hardy hierarchy | \(H_{\omega^22}(805)\) | \(H_{\omega^22}(806)\) |

Slow-growing hierarchy | \(g_{\omega^{\omega^{\omega^22+\omega4+5}}}(10)\) |

## Recalculation

Saibian later recalculated promaxima based on the idea that the universe would gradually vanish due to proton decay. Assuming that a proton has a half-life of about 10^{32} years, and using a rough estimate of 10^{81} protons in the universe, the universe could persist for at least 10^{35} years, a time period much longer than 500 quadrillion years. After factoring in the expansion rate of the universe (close to the speed of light) and changing the universal life span, promaxima reaches as high as 10^{10343}.^{[3]}

## Pronunciation

## Sources

- ↑ Really Big Numbers
- ↑ Sbiis Saibian's Ultimate Large Numbers List
- ↑ 2.1.7 - Larger Numbers in probability, Statistics, and Combinatorics

## See also

Sagan's number · Avogadro's number · Eddington number · Planck units · **Promaxima** · Poincaré recurrence time · Universe size