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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:

19923739028520154087706422945147014652916223529059455829739546236757445592829019852096549871643037231579555867729029727837739722687243833688041650758866703047684995147926044802500789969233229482277620428871361665114606086501621360310636409247822506979293012834235605892457887360583787492777424798206285182369042469497447438158240050711323245053205431372163355524614258748270064178183600550138767745559315784832858638844869498054620521042914198455705585134437206064557323165937735931605786380378378018264857422432758696743477636091751483267310595348292927018011128165226311150554708199087683524760666293693562405279021537

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[]

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

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea


Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system


Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory


Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud


See also: Template:Googology in Asia
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