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Mega

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For the SI prefix, see Mega-.

Mega is equal to Circle(2) or  in circle notation or Pentagon(2) in Steinhaus-Moser notation.[1] 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:

\[2[5] = 2[4][4] = 2[3]_2[4] = 2^{2}[3][4] = 4[3][4] = 4^{4}[4] = 256[4] = 256[3]_{256}\]

The last 14 digits computed by Sbiis Saibian are ...93539660742656.[2] More digits can be calculated quite easily using modular exponentiation.

It is the last number listed on Robert Munafo's Notable Properties of Specific Numbers.[3]

Values and approximations in other notations Edit

Matt Hudelson calls the number zelda.[4] In his version of Steinhaus-Moser notation, it is denoted Triangle(2).

Mega can be expressed as M(2,3) in Hyper-Moser notation[5] or \(2 \downarrow\downarrow\downarrow 259\) in down-arrow notation.

Mega can be bounded in arrow notation as:

\[10\uparrow\uparrow 257 < \text{Mega} < 10\uparrow\uparrow 258\ \text{or}\ 2\uparrow\uparrow 259 < \text{Mega} < 2\uparrow\uparrow 260\]

It can be bounded more precisely in Hyper-E notation:[2]

\[E1992373902852015408770642294514701465291622352905945582973954623675\\ 7445592829019852096549871643037231579555867729029727837739722687243\\ 8336880416507588667030476849951479260448025007899692332294822776204\\ 2887136166511460608650162136031063640924782250697929301283423560589\\ 2457887360583787492777424798206285182369042469497447438158240050711\\ 3232450532054313721633555246142587482700641781836005501387677455593\\ 1578483285863884486949805462052104291419845570558513443720606455732\\ 3165937735931605786380378378018264857422432758696743477636091751483\\ 2673105953482929270180111281652263111505547081990876835247606662936\\ 93562405279021537\#255\\ < \text{Mega} <\] \[E1992373902852015408770642294514701465291622352905945582973954623675\\ 7445592829019852096549871643037231579555867729029727837739722687243\\ 8336880416507588667030476849951479260448025007899692332294822776204\\ 2887136166511460608650162136031063640924782250697929301283423560589\\ 2457887360583787492777424798206285182369042469497447438158240050711\\ 3232450532054313721633555246142587482700641781836005501387677455593\\ 1578483285863884486949805462052104291419845570558513443720606455732\\ 3165937735931605786380378378018264857422432758696743477636091751483\\ 2673105953482929270180111281652263111505547081990876835247606662936\\ 93562405279021538\#255\]

It is therefore between giggol and giggolplex.

Mega is exactly equal to \(m(3)m(2)m(1)(2)\) in m(n) map.

Sources Edit

  1. Mega
  2. 2.0 2.1 3.2.5 - Mega - Large Numbers
  3. Notable Properties of Specific Numbers (page 22)
  4. Moser
  5. Large numbers by Aarex Tiaokhiao

See also Edit

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 · Flexipeta (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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