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Pampena's Prime is equal to $$10^{506}-10^{253}-1$$.[1] It is a a special type of prime number such that the entire prime is is formed with a concatenation of the same digit, save for a different digit in the middle of the number. It has gained notoriety due to the fact it was mentioned in a Numberphile video by mathematician Simon Pampena.

Decimal representation

The decimal representation of Pampena's Prime is as follows (with the 8 appearing in the middle bolded):

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Approximate size

Pampena's Prime is almost equal in size to one hundred censeptensexagintillion, or roughly a googol to the fifth power. In order to comprehend the scope of this number, one must construct a fifth dimensional hyper-cube such that each side has a googol smaller hyper-cubes, then multiply this construction by a million.

Notation Lower bound Upper bound
Scientific notation $$9.999\times10^{505}$$ $$1\times10^{506}$$
Arrow notation $$114\uparrow246$$ $$10\uparrow506$$
Steinhaus-Moser Notation 216[3] 217[3]
Copy notation 8[506] 9[506]
Taro's multivariable Ackermann function A(3,1677) A(3,1678)
Pound-Star Notation #*((69))*14 #*((70))*14
BEAF {114,246} {10,506}
Hyper-E notation 9E505 E506
Bashicu matrix system (0)(0)(0)(0)(0)(0)(0)[8976] (0)(0)(0)(0)(0)(0)(0)[8977]
Hyperfactorial array notation 255! 256!
Fast-growing hierarchy $$f_2(1670)$$ $$f_2(1671)$$
Hardy hierarchy $$H_{\omega^2}(1670)$$ $$H_{\omega^2}(1671)$$
Slow-growing hierarchy $$g_{\omega^{\omega2+86}53}(87)$$ $$g_{\omega^{\omega^25+6}}(10)$$

Sources

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