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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