- For the SI prefix, see Mega-.
Mega is equal to Circle(2) or ② in circle notation or Pentagon(2) in Steinhaus-Moser notation.[1][2] 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,[3] but the calculations are based on a wrong reasoning essentially explained in Tetration#Moduli_of_power_towers.[4] Saibian further calculated last 2048 digits.[5]
Fish calculated the last 10,000 digits on 14 July, 2022 and the last 100,000 digits on 14 October, 2023.[6]
Values and approximations in other notations[]
Matt Hudelson calls the number zelda.[7] In his version of Steinhaus-Moser notation, it is denoted Triangle(2).
Mega can be expressed as M(2,3) in Hyper-Moser notation[8], \(2 \downarrow\downarrow\downarrow 259\) 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.[3] \[E(n)\#255 < \text{Mega} < E(n+1)\#255\] 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 \(2^{f_2^{257}(2)}\).[9] 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.[10]
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 \(m_{n+1} = 256^{m_n} + m_n\).[11] 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.[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[]
- ↑ Hugo Steinhaus. Mathematical Snapshots Courier Corporation, 1999. ISBN 9780486409146 p.28
- ↑ Mega
- ↑ 3.0 3.1 Large Numbers - 3.2.5 - Mega
- ↑ Talk:Mega#On_Saibian's_computation
- ↑ mega_storage
- ↑ Fish Last 100000 digits of mega 2023-10-14
- ↑ Moser
- ↑ Large numbers by Aarex Tiaokhiao
- ↑ Talk:Mega#Fast-growing_hierarchy
- ↑ Fish. Approximation of FGH with arrow notation 2023-12-05.
- ↑ A difference page of the talk page of this article.
- ↑ Robert Munafo. Source code of Hypercalc Retrieved 2023-12-12.
See also[]
Mega series: Mega · A-ooga (Megision) · Megisiduon · Megisitruon · Megisiquadruon
Grand Mega series: Grand Mega · Grand Megision · Grand Megisiduon (A-oogra) · Grand Megisitruon · Grand Megisiquadruon
Great Mega series: Great Mega · Great Megision · Great Megisiduon · Great Megisitruon (A-oogrea) · Great Megisiquadruon
Gong Mega series: Gong Mega · Gong Megision · Gong Megisiduon · Gong Megisitruon · Gong Megisiquadruon (A-oogonga)
Hexomega series: Hexomega · Hexomegision · Hexomegisiduon · Hexomegisitruon · Hexomegisiquadruon · Hexomegisiquinton (A-oohexa)
Heptomega series: Heptomega · A-oohepta (Heptomegisisexton) · Octomega · A-oocta · Nonomega · A-ooennea
Megistron series: Megiston (Megistron) · Megisiplextron · Megisiduplextron · Megisitriplextron · Megisiquadruplextron · A-oomega (Megisienneaplextron)
A-ooga series: A-ooga · Betomega (A-oogatiplex) · A-oogatiduplex · A-oogatitriplex · A-oogatiquadruplex · A-oogatiquintiplex
A-oogra series: A-oogra · A-oogratiplex · Betogiga (A-oogratiduplex) · A-oogratitriplex · A-oogratiquadruplex · A-oogratiquintiplex
A-oogrea series: A-oogrea · A-oogreatiplex · A-oogreatiduplex · Betotera (A-oogreatitriplex) · A-oogreatiquadruplex · A-oogreatiquintiplex
A-oogonga series: A-oogonga · A-oogongatiplex · A-oogongatiduplex · A-oogongatitriplex · Betopeta (A-oogongatiquadruplex) · A-oogongatiquintiplex
A-oohexa series: A-oohexa · A-oohexatiplex · A-oohexatiduplex · A-oohexatitriplex · A-oohexatiquadruplex · Betoexa (A-oohexatiquintiplex)
A-oohepta series: A-oohepta · Betozetta (A-ooheptatisextiplex) · A-oocta · Betoyotta · A-ooennea · Betoxota
A-oomega series: A-oomega · A-oomegatiplex · A-oomegatiduplex · A-oomegatitriplex · A-oomegatiquadruplex · Betodaka (A-oomegatienneaplex)
Betomega series: Betomega · Flexinega (Brantomega) · Breatomega · Bigiatomega · Biquadriatomega · Biquintiatomega
Betogiga series: Betogiga · Brantogiga · Flexitria (Breatogiga) · Bigiatogiga · Biquadriatogiga · Biquintiatogiga
Betotera series: Betotera · Brantotera · Breatotera · Flexitera (Bigiatotera) · Biquadriatotera · Biquintiatotera
Betopeta series: Betopeta · Brantopeta · Breatopeta · Bigatopeta · Flexipera (Biquadriatopeta) · Biquintiatopeta
Betoexa series: Betoexa · Brantoexa · Breatoexa · Bigatoexa · Biquadriatoexa · Flexiexa (Biquintiatoexa)
Betozetta series: Betozetta · Flexizetta · Betoyotta · Betoxota · Betodaka
Flexinega series: Flexinega · Oktia (Fainega) · Funnynega · Ftetrinega · Fpentinega · Fhexinega
Flexitria series: Flexitria · Faitria · Oktria (Funnytria) · Ftetritria · Fpentitria · Fhexitria
Flexitera series: Flexitera · Faitera · Funnytera · Oktetra (Ftetritera)
Moser series: Moser · Grand Moser · Great Moser · Gong Moser
Maser series: Maser · Miser (Killaser) · Meser · Muser
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