Belphegor's prime is the name for the number 1,000,000,000,000,066,600,000,000,000,001 = \(10^{30} + 666 \cdot 10^{14} + 1\).[1] It is a palindromic prime, notable for its curious decimal expansion containing the beast number. Clifford Pickover named the number after a demon from the Judeo-Christian mythos, playing on 666's association with the devil in numerology.
In terms of digit repetition, the number could be written \(1\underbrace{000\ldots000}_{13}666\underbrace{000\ldots000}_{13}1\). (The presence of the number 13 — associated with bad luck in Western culture — further ties into the numerological theme.)
The Belphegor's primes may also be defined as prime numbers of the form \[B_n = 10^{2n+4} + 666 \cdot 10^{n+1} + 1 = 1\underbrace{000\ldots000}_n666\underbrace{000\ldots000}_n1\] and the first few Belphegor's primes are \(B_n\) for \(n = 0,\,13,\,42,\,506,\,608,\,2472,\,2623,\,28291,\,181298\).[2][3]
Approximations[]
Notation | Lower bound | Upper bound |
---|---|---|
Scientific notation | \(1\times10^{30}\) | \(1.001\times10^{30}\) |
Arrow notation | \(10\uparrow30\) | \(75\uparrow16\) |
Steinhaus-Moser Notation | 22[3] | 23[3] |
Copy notation | 9[30] | 1[31] |
Taro's multivariable Ackermann function | A(3,96) | A(3,97) |
Pound-Star Notation | #*(9,6,4)*9 | #*(0,1,0,0,1)*5 |
BEAF | {10,30} | {75,16} |
Hyper-E notation | E30 | E[75]16 |
Bashicu matrix system | (0)(0)(0)[5623] | (0)(0)(0)[5624] |
Hyperfactorial array notation | 28! | 29! |
Fast-growing hierarchy | \(f_2(93)\) | \(f_2(94)\) |
Hardy hierarchy | \(H_{\omega^2}(93)\) | \(H_{\omega^2}(94)\) |
Slow-growing hierarchy | \(g_{\omega^{\omega3}}(10)\) | \(g_{\omega^{\omega+8}12+\omega^{\omega+7}10}(16)\) |
Sources[]
- ↑ Clifford Pickover, Belphegor's Prime: 1000000000000066600000000000001!
- ↑ MathWorld, Belphegor Prime
- ↑ OEIS A232448
See also[]
Specific numbers: Beast number · Belphegor's prime · Hyper-leviathan number · Legion's number of the first kind · Legion's number of the second kind · Leviathan number
Forms of numbers: apocalypse number · apocalyptic number · apocalyptri number · apocalyptetra number · apocalypenta number · apocalyhexakosioihexekontahexa number