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Belphegor's prime is the name for the number 1,000,000,000,000,066,600,000,000,000,001 = $$10^{30} + 666 \cdot 10^{14} + 1$$. 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.) Harvey Dubner calculated that the first few prime numbers of the form $$1\underbrace{000\ldots000}_n666\underbrace{000\ldots000}_n1$$ have $$n = 0,\,13,\,42,\,506,\,608,\,2472,\,2623$$[citation needed].

## 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 23
Copy notation 9 1
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 E16
Bashicu matrix system (0)(0)(0) (0)(0)(0)
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 $$g_{\omega^{\omega3}}(10)$$ $$g_{\omega^{\omega+8}12+\omega^{\omega+7}10}(16)$$