Googology

Googology is the study of numbers between a googol and a googolplex and higher.

Here are some examples of such numbers using the Ackermann’s Generalized Exponential notation, g(a, b, c) = g(a - 1, g(a, b - 1, c), c), g(0, a, b) = a + b, g(1, a, b) = ab:

g(2, 100, 10) = ten-to-the-hundredth =

g(2, 2, g(3, 2, 10)) = to-the-second-ten squared =

g(2, 50, g(1, 2, 50)) = twice-fifty-to-the-fiftieth.

great googol = g(2, 3/2, g(2, 100, 10)) = g(2, 1000, 10)

googolteen = g(2, 100, 10) + 10

googolty = g(2, 101, 10)

googolplex = g(2, g(2, 100, 10), 10) = g(2, g(2, g(2, 2, 10), 10), 10) = g(3, 2, 2, 10)

eleventyplex = g(2, 110, 10)

googolex = g(2, 60, 120) > 5g(2, 124, 10)

googoc = g(2, 100, 200) > g(2, 230, 10)

googoci = g(2, 101, 202) > 6g(2, 232, 10)

googocci = g(2, 201, 402) > 2g(2, 523, 10)

googoccic = g(2, 299, 598) > g(2, 830, 10)

googoxem = g(2, 990, 1980) >5g(2, 3263, 10)

googolmox = g(2, 1003, 2006) > g(2, 3312, 10)

googomump = g(2, 2000, 4000) > g(2, 7204, 10)

googolplox = g(2, g(2, 50, 100), 3) > g(2, 2.002, g(3, 3, 10))

googolbar = g(2, 50000, 100000) = g(2, 250000, 10)

googocbar = g(2, 100000, 200000) > 5g(2, 530103, 10)

googodbar = g(2, 500000, 1000000) = g(2, 3000000, 10)

googombar = g(2, 1000000, 2000000) > 9g(2, 6301029, 10)

googomembar = g(2, 2000000, 4000000) > 9g(2, 13204119, 10)

umpbarbar = g(1, g(2, 1000, 1000), 1000) = g(3, 2, 1000) + 3, 10) = g(2, 3003, 10)

googolplexian = g(2, g(2, g(2, g(2, 2, 10), 10), 10), 10) = g(4, 1, 2, 2, 10)

googolplux = g(g(2, 100, 10), g(2, 100, 10), 1, 1, 2, 100, 10) = g(g(3, 2, g(2, 100, 10)), 1, 1, 2, 100, 10)

Which extrapolates, since c = g(2, 2, x) and m = g(2, 3, x) to:

googolpluc = g(g(g(2, 2, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)

googolplum = g(g(g(3, 3, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)

baggoogol = g(2, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

traggoogol = g(3, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

quadraggoogol = g(4, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

centaggoogol = g(100, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

dentaggoogol = g(500, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

mentaggoogol = g(1000, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

bentaggoogol = g(g(1, 2, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

trentaggoogol = g(g(1, 3, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))

yottaggoogol = g(g(2, 80, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))