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.

Using exponents to represent digit repetition, the number could be written \(10^{13}6660^{13}1\). (The presence of the number 13 — associated with bad luck in Western culture — further ties into the numerological theme.) Harvey Dubner calculated that the first few numbers of the form \(10^n6660^n1\) have \(n = 0,\,13,\,42,\,506,\,608,\,2472,\,2623\).

Approximations Edit

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 Edit

  1. Clifford Pickover, Belphegor's Prime: 1000000000000066600000000000001!

See also Edit