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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!