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Grangol (short for "grand googol") is a number equal to a power tower of 100 tens topped with a 100.[1][2] It can be defined in Hyper-E Notation as E100#100 = EEE...EEE100 (100 E's) = 1010...10100 (100 tens). The term was coined by Sbiis Saibian.

Grangol is comparable to 10 tetrated to 101, using hyper-operators.

Aarex Tiaokhiao calls this number powergol.[3]

Comparison with giggol

Grangol is close to and slightly larger than Jonathan Bowers' giggol = E1#100. It is comparable only from a googologist's perspective, however — it can be shown that:

$$\text{giggol}^{\text{giggol}}$$$$< \text{grangol}$$

so the two numbers are arithmetically extremely far apart. This concept is related to the power-tower paradox.

The proof is fairly straightforward. Since giggol = E1#100 and Ed#p = 10Ed#(p-1) it follows that:

\begin{eqnarray*} \text{giggol}^{\text{giggol}} &=& E1\#100^{E1\#100} = (10^{E1\#99})^{E1\#100} \\ &<& 10^{(E1\#100)^2} = 10^{10^{2E1\#99}} = E(2E1\#99)\#2 \\ &<& E(10E1\#99)\#2 = E(10\times10^{E1\#98})\#2 \\ &=& E(10^{1+E1\#98})\#2 = E(1+E1\#98)\#3 \\ &<& E(1+E1\#97)\#4 < \cdots < E(1+E1\#1)\#100 \\ &=& E(1+E1)\#100 = E11\#100 \\ &<& E100\#100 = \text{grangol} \end{eqnarray*}

Therefore, giggolgiggol can be given the following lower and upper bound:

$E10\#100 < \text{giggol}^{\text{giggol}} < E11\#100$

Approximations in other notations

Notation Approximation
Up-arrow notation $$10 \uparrow\uparrow 101$$
Chained arrow notation $$10 \rightarrow 101 \rightarrow 2$$
BEAF $$\{10,101,2\}$$
Hyperfactorial array notation $$104!1$$
Strong array notation $$s(10,101,2)$$
Nested factorial notation $$100![2]$$
Fast-growing hierarchy $$f_3(100)$$
Hardy hierarchy $$H_{\omega^3}(100)$$
Slow-growing hierarchy $$g_{\varepsilon_0}(101)$$