Mega

Mega is equal to Circle(2) or ② in circle notation or Pentagon(2) in Steinhaus-Moser notation.Hugo Steinhaus. Mathematical Snapshots Courier Corporation, 1999. ISBN 9780486409146 p.28Mega It was defined by Hugo Steinhaus along with the megiston in the book Mathematical Snapshots. Mega can also be defined recursively as \(m_{256}\) in the sequence defined by \(m_0 = 256\) and \(m_{n + 1} = m_n^{m_n}\).
Steinhaus showed that it is equal to Square(256):
:Pentagon(2) = Square(Square(2)) = Square(Triangle(Triangle(2))) = Square(Triangle(4)) = Square(256) = Triangle256(256)
Using the general notation proposed by Susan Stepney, mega is:
\(2[5] = 2[4][4] = 2[3]_2[4] = 2^{2}[3][4] = 4[3][4] = 4^{4}[4] = 256[4] = 256[3]_{256}\)
== Last digits ==
The last 14 digits computed by a program of Sbiis Saibian are ...93,539,660,742,656,Large Numbers - 3.2.5 - Mega but the calculations are based on a wrong reasoning essentially explained in Tetration#Moduli_of_power_towers.Talk:Mega#On_Saibian's_computation Saibian further calculated last 2048 digits.mega_storage
Fish calculated the last 10,000 digits on 14 July, 2022 and the last 100,000 digits on 14 October, 2023.Fish Last 100000 digits of mega 2023-10-14
== Values and approximations in other notations ==
Matt Hudelson calls the number zelda.Moser In his version of Steinhaus-Moser notation, it is denoted Triangle(2).
Mega can be expressed as M(2,3) in Hyper-Moser notationLarge numbers by Aarex Tiaokhiao, \(2 \downarrow\downarrow\downarrow 259\) in down-arrow notation, or 2[2] in ampersand notation.
Mega can be bounded in arrow notation as:
\[10\uparrow\uparrow 257 < \text{Mega} < 10\uparrow\uparrow 258\]
or
\[2\uparrow\uparrow 259 < \text{Mega} < 2\uparrow\uparrow 260\]
by using linear approximation of tetration,
\[\text{Mega} \approx 10 \uparrow\uparrow 257.4458999580817\]
It can be bounded more precisely in Hyper-E notation.
\[E(n)\#255 < \text{Mega} < E(n+1)\#255\]
where n is equal to:
therefore
\[E(619)\#256 < \text{Mega} < E(620)\#256\]
or
\[\text{Mega} \approx E(619.29937084448)\#256 \approx E(2.7919006388035)\#257\]
It is therefore between giggol and giggolplex.
Mega is exactly equal to \(m(3)m(2)m(1)(2)\) in m(n) map because
* \(m(1)(n)=n[3]\)
* \(m(2)m(1)(n)=n[4]\)
* \(m(2)^{p-3}m(1)(n)=n[p]\)
* \(m(3)m(2)m(1)(n)=n[n+3]\)
* \(m(3)m(2)m(1)(2)=2[5]\)
Mega is very closely upper bounded by \(f_2^{258}(2)\) in the fast-growing hierarchy, and is exactly equal to \(2^{f_2^{257}(2)}\).Talk:Mega#Fast-growing_hierarchy This is due to the laws of exponents: \({(2^n)}^{(2^n)} = 2^{n \times 2^n} = 2^{f_2(n)} \). Simply observe:
* 256 = \(2^8\)
* \(256^{256}\) = \({(2^8)}^{(2^8)}\) = \(2^{8\times{2^8}}\) = \(2^{2^{11}}\) = \(2^{2048}\) = \(2^{f_2^2(2)}\) 
* 256[3][3] = \(2^{2048}[3]\) = \({(2^{2048})}^{(2^{2048})}\) = \(2^{2048\times(2^{2048})}\) = \(2^{2^{2059}}\) = \(2^{f_2^3(2)}\)
Mega can also be approximated by \(f_3(256)\), which is approximately \(10 \uparrow\uparrow 257.27814860577\) by using Fish's program.Fish. Approximation of FGH with arrow notation 2023-12-05.
A Googology Wiki user Tetramur pointed out that Mega can be exactly expressed via single power-tower as \(256^{256^{m_{256}}}\) in the sequence \(m_0 = 0\) and \(m_{n+1} = 256^{m_n} + m_n\).A difference page of the talk page of this article. Since there are two \(m_n\) in the recursive formula, the size of expression is doubled each time. Therefore, it is impossible to write down entire power tower since it contains no less than \(2^{256}\) symbols.
== Program ==
Robert Munafo writes a program to calculate mega with BASIC program implemented in Hypercalc in its help page as follows.Robert Munafo. Source code of Hypercalc Retrieved 2023-12-12.
 5 ' Calculate Hugo Steinhaus' number "Mega"
 10 let mega=256;
 20 for n=1 to 256;
 40 let mega = mega ^ mega;
 80 next n
 160 print "Mega = "; mega
 320 end
== Sources ==
== See also ==
*A-ooga
*Circle notation
*Megiston
*Moser
*Steinhaus-Moser Notation
ja:メガ
zh:美卡
Category:Numbers
Category:Tetration level
Category:Hyper-Moser numbers
Category:Powers of 2
Category:Numbers with peer-reviewed source