# Historical records

People used to take large numbers seriously, going to great lengths to find larger ones, I suppose. We examine the history of Big number records – from the ancient Greeks to a Cheltenham asperger with a degree in maths.

# Archimedes octads

The history of Big numbers starts with a letter by Archimedes in the 3d century BC sent to King Gelon β of Syracuse. Popularly known as the Sand Reckoner, Archimedes' letter calculates the number of sand grains that “the sphere of the fixed stars” can contain. Archimedes gives an estimate of at most 10^63 = 1E63 sand grains, expressed with the octads (myriad myriads ordered in periods), which he had introduced in an earlier mathematical treatise.

Before the Greeks named 1E8 as a myriad myriads. Try to count a googol 1E1E2 with that! Archimedes would say a googol equals “a myriad of octads of the 13th order”. But the googolplex 1E1E100 lies out of reach for Octads.

The last number in the Octads system 10^(8*10^16) = 1E8E16 would have been the ancient Greek number record.
His system's expressive power leads Archimedes to the conclusion that the numbers constructible in mathematics are much larger than any quantity in the physics of the universe. And thus he puts his King's mind to rest.

# Buddhist unspeakable tower

End 7th century AD the Chinese empress Wu commissioned buddhist monks with the translation of the Avatamsaka Sutra from Sanskrit. In this magnificent scripture the empress would have read a description of the largest quantity ever conceived by man.

The translation by Siksananda has the initial number of a hundred laksha 10^7 squared 103 times till the incalculable asamkhyeya 10^(7*2^103) is observed (near the auspicious 2^2^108 and 10^10^32 power towers). Then by double squaring it reaches the unspeakable 10^(7*2^119) and the untold 10^(7*2^121) – numbers used in the following poem in the Avatamsaka (Thomas Cleary, "Flower Ornament Scripture", chapter 30, page 892).

The atoms [containing untold lands] to which these [atomizable] buddha-lands are reduced in an instant are unspeakable, and so [atom to 10^(35*2^119) atoms] are the atoms of continuous reduction moment to moment, going on for untold eons… [up to 10^10^2^124 atoms]

Counting this way [power upon power] for unspeakable eons, using unspeakable numbers, counting eons [recursively] by these atoms…
[Siksananda's total 10^^(10^(7*2^119)) atoms | lands | eons] 

These verses express a number of about 10^..1 :2^2^124 by means of a power tower of unspeakable size. This record number from the literature of ancient India and China is the first example of tetration in the history of mathematics.

# Guinness Graham's number

David Hilbert in On the Infinite (1925) proposed that a double recursion (his algorithm equals our superpower stars) increases faster than any primitive recursive function. This was proven by Wilhelm Ackermann in his 1928 article, and such functions (over the superpower index) are since called Ackermann functions.

In 1927 the Dutch mathematician Bartel van der Waerden founded the theory of arithmetical progressions in Ramsey Theory. Upper bounds for his numbers were once estimated using double recursion (but nowadays by a power 2^k#6 tower).
Ronald Graham is an expert of general Ramsey Theory, where the limits can be pushed a lot further. He came up with this huge number, to illustrate a powerful theory…

Martin Gardner's 1977 version of Graham's number once featured in the Guinness book of World Records as “the highest number ever used in a mathematical proof”. This record number counts 64 layers of up-arrows (double recursions), and is famously called “Graham's number”.

Graham's number is an upper bound for the nth dimension, where connecting points (vertices) in a 2-coloured hypercube forces a certain progression of colours (on the edges). The exact dimension where such a progression must exist is probably quite low, but while computers paint hypercubes at random up to exhaustion, a solution is still not found.

But the final number of Graham & Rothschild's original 1971 article has just 7 double recursions nested (of a special Ackermann function). Although their generalized formula seems capable of producing much larger upper bounds (for more exotic Ramsey problems than to solve a certain progression in a 2-coloured hypercube).

We will express both record numbers by nesting :n: chained arrows to their respective depth. Appending #n in a Hyper expression achieves the same iteration.

2→3→(..12..) :7: = 2→3→12#7  (original Graham's)
3→3→(..4..) :64: = 3→3→4#64  (famous Graham's)

It is awkward to accept the latter number as a record, because it resulted from a miscalculation (bake the blame). The earlier bound by the enigmatic Ron Graham is way too high already!
It was lowered by Saharon Shelah and now a proper upper bound (for the solution of a Hales-Jewett hypercube) can generally be expressed by some 2→k→5 class function.
Higher bounds for Ramsey problems are to be expected from certain sets of complete subhypergraphs. We question if any colour recipe for combined graphs requires Ackermann functions to express its upper bound, like Graham's hypercube problem used to do.

# Bowers meameamealokkapoowa oompa

A clumsily defined number record from a Big number enthusiast from Tyler, Texas.
The meameamealokkapoowa numbers were coined by Jonathan Bowers at the end of his explosive list of names for Infinity Scrapers.
But we lose track on the way he gets there, at the Legion Legions structures – wild ideas that resulted from his email sessions with John Spencer. For us Bowers' Infinity Scrapers are still acceptable somewhere in the Legions, it's there the trail becomes misty.

High up in his number tree Hedrondude Bowers loses sight of his prime invention – the upload methods – when he places novel systems on top of the old. By nesting higher structures to the right, he severs the connection with the engine of his airplane – in effect making a fresh start from that point on.
Best is to let all structures increase (explode) from the prime entry below – by upload of its value to function as an index for the new structures. This should be well-defined – specially at the moment of the jump, the parachute should open ;-)

The third man involved in this record was Chris Bird, who on his own supplied a definition for the Meameamealokkapoowa Oompa, which requires a study of Bird's system to begin with.
Chris tells us his array notations easily topple all of Bowers' structures. We believe him, Chris Bird is the ruling world champion, the man of Big numbers to beat.

# Bird beyond nested arrays

At present the best algorithm to construct the largest natural numbers, is the hyper-nested arrays of Christopher M. Bird from Cheltenham in England. Bird's arrays were originally inspired by the upload rule and hyper-dimensional structures of Jabe Bowers, who coined the term Beaf. But when we write Beaf you should think “Bird's Extended Array Functions” – Bird defined these algorithms and extended the array structures properly.

Greetings from The Hague, Holland! We are keen on designing a Btrix matrix that improves on Bird III – the best system of Chris Bird – his Beyond Nested Arrays III.
Our goal is to create a new system, that runs rule-significantly faster than the final function in Bird III, the current world record holder H:

H(..3..) :H(3):  (eternal laughter)
              = H(3#1#2 = H(3#H(3)

The former expression applies our RepExp notation. Two words are repeated from both ends of the expression inwards up to the double dots .. in tandem :k: times.
The latter expression is equal, but written with Hyper hashes # and single ( brackets.

In this article we'll continue to measure our Btrix matrix against Bird's system on the fly. If we can manage to jump from the nest and spread our wings, an exact record Máxima number can surely be constructed high in the sky.

# Rayo afterwords

We feel the record number of the logician Agustín Rayo, “the smallest number bigger than any finite number named by an expression in the language of first order set theory with a googol symbols or less”, apart from its fundamental failure to result in an exact number, is contrived in its motivation, and (surprise?) far from optimal.

The second order definition of Rayo's number by Gödel-alphabetization overlooks the true mechanism for creating Big numbers. Why?

Any expression can be translated to a simpler expression with less symbols and yet in the end be reduced to a larger number 1... if only the rule system is Big enough. So the size and complexity of the Turing machine itself should be taken into account, not just the sizes of the tape it reads.
It is the typification of symbols itself that we seek to subject to recursion. “Symbol” is too fleeting a concept in the context of Big number algorithms.

Eventually we'd like to upload the size of the alphabets and the indices of Turing machines way back from below. We think interconnecting all these virtual machines leads to mind-bigglingly Bøg numbers. Rayo can keep his set-theoretic universes separate, but these stay mighty inefficient compared to running an ongoing recursion in between. Why?

Compare the upload system of Bowers & Bird with the archetypical download systems – the Hyper iterations of Sbiis always stay on top, and Conway's chained arrows never replace higher entries by previously incremented lower entries. Given the same alphabet size and symbolic complexity downloads result in Bigger numbers than uploads.
The down and dirty upload of Beaf, with its exploding higher array structures, gains speed (this is a conjecture to prove). It will one day upload new signs to alphabets and climb the hierarchies of hyperarithmetical Turing machines…

by Giga Gerard

People can read this article plain or choose the hard & fast & open version.