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Grangol in zeroes

Grangol in towers

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.

This is a 1 followed by E100#99 zeros, or a googol-99-plex

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 \(g_{\varepsilon_0}(101)\)

Sources

See also

Gugold — Throogol
Gugold—gugolthra series: gugold · gugoldagong · great googol ·gugolda-suplex · gugolda-suplexigong · gugolda-dusuplex · gugolda-dusuplexigong · graatagold · graatagoldagong · graatagolda-sudex · graatagolda-sudexigong · graatagolda-dusudex · graatagolda-dusudexigong · greegold · greegoldagong · greegolda-suthrex · greegolda-suthrexigong · greegolda-dusuthrex · greegolda-dusuthrexigong · grinningold · grinningoldagong · grinningolda-sutetrex · grinningolda-sutetrexigong · grinningolda-dusutetrex · grinningolda-dusutetrexigong · golaagold · golaagoldagong · golaagolda-supentex · golaagolda-supentexigong · golaagolda-dusupentex · golaagolda-dusupentexigong · gruelohgold · gruelohgoldagong · gruelohgolda-suhex · gruelohgolda-suhexigong · gruelohgolda-dusuhex · gruelohgolda-dusuhexigong · gaspgold · gaspgoldagong · gaspgolda-suheptex · gaspgolda-suheptexigong · gaspgolda-dusuheptex · gaspgolda-dusuheptexigong · ginorgold · ginorgoldagong · ginorgolda-suoctex · ginorgolda-suoctexigong · ginorgolda-dusuoctex · ginorgolda-dusuoctexigong · gargantuuld · googondold
Gugolthra—throogol series: gugolthra · graatagolthra · greegolthra · grinningolthra · golaagolthra · gruelohgolthra · gaspgolthra · ginorgolthra · gargantuulthra · googondolthra · gugoltesla · graatagoltesla · greegoltesla · grinningoltesla · golaagoltesla · gruelohgoltesla · gaspgoltesla · ginorgoltesla · gargantuultesla · googondoltesla · gugolpeta · graatagolpeta · greegolpeta · grinningolpeta · golaagolpeta · gruelohgolpeta · gaspgolpeta · ginorgolpeta · gargantuulpeta · googondolpeta · gugolhexa · graatagolhexa · greegolhexa · grinningolhexa · golaagolhexa · gruelohgolhexa · gaspgolhexa · ginorgolhexa · gargantuulhexa · googondolhexa · gugolhepta · graatagolhepta · greegolhepta · grinningolhepta · golaagolhepta · gruelohgolhepta · gaspgolhepta · ginorgolhepta · gargantuulhepta · googondolhepta · gugolocta · graatagolocta · greegolocta · grinningolocta · golaagolocta · gruelohgolocta · gaspgolocta · ginorgolocta · gargantuulocta · googondolocta · gugolenna · graatagolenna · greegolenna · grinningolenna · golaagolenna · gruelohgolenna · gaspgolenna · ginorgolenna · gargantuulenna · googondolenna · gugoldeka · graatagoldeka · greegoldeka · grinningoldeka · golaagoldeka · gruelohgoldeka · gaspgoldeka · ginorgoldeka · gargantuuldeka · googondoldeka
Throogol — Godgahlah-ex-grand godgahlah
Throogol—tetroogol series: throogol · thrangol · threagol · thrigangol · throrgegol · thrulgol · thraspgol · thrinorgol · thrarantuul · throogondol · thrugold · thraatagold · threegold · thrinningold · throlaagold · thruelohgold · thraspgold · thrinorgold · thrargantuuld · throogondold · thrugolthra · thrugoltesla · thrugolpeta · thrugolhexa · thrugolhepta · thrugolocta · throotrigol · thrantrigol · threatrigol · thrutrigold · thraatatrigold · threetrigold · thrutrigolthra · thraatatrigolthra · thrutrigoltesla · thrutrigolpeta · thrutrigolhexa · throotergol · thrantergol · threatergol · thrutergold · thraatatergold · thrutergolthra · thrutergoltesla · thrutergolpeta · thrutergolhexa · throopetol · thranpetol · thrupetold · thrupetolthra · throohexol · thranhexol · thruhexold · thruhexolthra · throoheptgol · throogogdol · throogentol · throodekol
Tetroogol—grand grand godgahlahgong series: tetroogol · tetrangol · tetreagol · tetrugold · tetraatagold · tetrugolthra · tetrugoltesla · tetrithroogol · tetrithrootrigol · tetrithrootergol · tetrithroopetol · tetrithroohexol · tetrootrigol · tetratrithroogol · tetratrithrootrigol · tetrootergol · tetraterithroogol · tetroopetol · tetroohexol · tetrooheptgol · tetroogogdol · tetroogentol · tetroodekol · pentoogol · hexoogol · heptoogol · ogdoogol · ogdahepti-hexapenti-tetrithroogol · ogdatri-hexi-pentateri-tetrapetithrinortrigolthra · entoogol · dektoogol
Godgahlah - godgahlah-ex-grand godgahlah series: godgahlah · godgahlahgong · grand godgahlah · grand godgahlahgong · grand grand godgahlah · grand grand grand godgahlah · grand grand grand grand godgahlah · hundred-ex-grand godgahlah · godgahlah-ex-grand godgahlah

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