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SI Prefixes
The original SI (systeme international, or "Metric" system) in 1793 had only the prefixes kilo, hecto, deca, deci, centi, and milli. In 1795 myria was added, but it was not included in the international system established in 1960, and therefore became obsolete. mega dates from the late 1800's and was officially adopted in France in 1919. During the 1900's compound prefixes kilomega and megamega were used, and it was eventually decided these needed their own prefixes3; for this purpose giga, nano and tera, pico were part of the international system adopted in 1960. The system has gradually expanded as more applications arise, and this includes new prefixes: femto and atto in 1964, peta and exa in 1975, zetta, zepto, yotta, and yocto in 1991, and ronna, ronto, quetta, and quecto in 2022.
These extensions have been added mainly for the convenience of scientists. For example, in 1993 some researchers had to refer to units of 10-21 volts but they didn't yet know about the prefix zepto so they called it "milliattovolt". There have always been fields in which very small or very large values are expressed simply as powers of 10, or using a field-specific and somewhat arbitrary unit — examples include the parsec, electronvolt, and angstrom. The standard unit of atomic mass (1/12 the mass of a Carbon atom, or roughly the mass of a proton or a neutron) is 1.66×10-27 kg or 1.66 yg (1.66 yoctograms). Quasars have been discovered that are about 125 yottameters away. When volumes or weights are involved, more examples can be found that exceed the range provided by the standard prefixes. Until 2022 one needed to say that the Earth weighs about 6000 Yg (6000 yottagrams), and the mass of an electron is about 0.00091 yg (yoctograms); but now these are the easier-to-say *6 ronnagrams and 910 quectograms" respectively. By comparison, the diameter of the Earth and of an atom (both one-dimensional measurements) are easily handled with the older kilo- and pico- prefixes.
The official SI prefix names "peta" and "exa" are probably derived from "pente" and "hexa" (five and six in Greek)3, however, the official BIPM (Bureau International des Poids at Mesures) website does not give an explanation.
"Zetta" and "yotta" are derived from "septo" and "octo", which are quasi- number names. Quoting BIPM:13
The names zepto and zetta are derived from septo suggesting the number seven (the seventh power of 103) and the letter "z" is substituted for the letter "s" to avoid the duplicate use of the letter "s" as a symbol. The names yocto and yotta are derived from octo, suggesting the number eight (the eighth power of 103); the letter "y" is added to avoid the use of the letter "o" as a symbol because it may be confused with the number zero. — 13
The published report43 on the meeting that produced these prefixes also states, "In preparing ils proposai, the CCU tried to respect the following rules: a) prefixes should end in 'o' for negative powers and 'a' for positive powers; h) symbols for prefixes should be capital letters for positive powers and lower case letters for negative powers." There is no official statement regarding the choice of the letters z and y, though many unofficial proposals, discussed below, took the hint that more letters from the end of the alphabet could be used.
For some time there was speculation about future BIPM prefixes for 1027 and 1030 and their reciprocals 10-27 and 10-30. I found suggestions that they might resemble names for nine and ten e.g. novetta, novemo, decetta and decemo. A reason for not doing this is that their abbreviations would almost certainly need to have two letters, because N-, n- and d- are already used for other prefixes. Many other speculative ideas used letters near the end of the alphabet: x-, w-, v-, and so on, but again there is clearly a conflict with units such as W for watt and V for volt.
The actual prefixes added in 2022 use new letters R and Q. A BIPM working document from 2019 confirms42 the concern for avoiding letters already in use and explains that "ronna" and "quetta" (originally "quecca") do indeed derive from nine and ten in other languages:
Letters already in use for SI prefixes, SI units, other common units, or symbols that may otherwise cause confusion, should be avoided. Following the precedent set recently, letters should be used in reverse English alphabetical order, suitably modifying chosen names, and skipping letters as appropriate. [...] "ronto" [is from] Greek & Latin, derived from ‘ennea’ and ‘novem’, suggesting 9; [...] "quecto" [from] Latin, derived from ‘decem’, suggesting 10; [and "ronna" and "quecca" for the same reasons: ["nona" with the n changed to r is "rona", and "deka" with the d changed to qu is "queka"]] — 42
The same source suggests that 'b' is the only suitable letter remaining, and speculates on bundecca and bundecto (probably based on Latin "undecim") for 1033 and 10-33.
Joke, Hoax, and Sincere (but Ad-Hoc) SI Prefix Proposals
The following prefixes have all been exposed as jokes or hoaxes: bronto, cuppa, dea, ento, fito, harpo, hepa, lotta, lotto, nea, otta, revo, syto, tredo, una, xera, zeppo, and zuppa3,27,28,29,33,34. Several of these were perpetuated by Internet rumors (compare two posts by Alex Lopez-Ortiz 4,11), and many go back before widespread use of the Internet12.
One of the earliest and best-known examples are the "Marx Brothers prefixes", apparently a reaction to the silly-sounding and real prefixes yocto and zepto. In 1993, in a joke that received "general approval on Usenet", Morgan Burke27 proposed harpo for 10-27 and groucho for 10-30 (and therefore harpi for 1027 and grouchi for 1030)29.
In addition, the following are simply bogus ad-hoc personal ideas: luma, lunto, mikto, minga, nekto, nena, ocha, otro, pekro, pepta, quekto, quexa, rimto, rinta, sorta, sotro, treda, trekto, uda, udeka, udeko, unto, vendeka, vendeko, vunda, vunkto, weka, weko, wekta, wekto, xenno, xenta, xona, and xonto10.
A much-repeated anecdote claims that Oxford professor Jeff K. Aronson (who is indeed a real person) suggested extending beyond zetta, zepto and yotta, yocto with xenta=1027, xenno, wekta=1030, weko, vendeka=1033, vendeko, udeka=1036, and udeko.
I could not verify Dr. Aronson as a source; instead I found several mentions of the xenta/xenno and vendeka/vendeko pairs (but without the w- and u- pairs) on USENET during the early and mid 1990's, crediting New Scientist magazine44 as their source. Unfortunately, this turns out to be from the magazine's weekly "Last Word" page (where readers answer each others' questions). Thus, the claim that "Others such as xenno (x) (10−27) and xenna (X) (1027), or vendeko (v) (10−33) and vendeka (V) (10−33) exist [...]" was simply made by another reader. These appear for example in the earliest revisions of a physics textbook by Christoph Schiller47, which later48 added the "Aronson" w- and u- pairs, and inserted extra letters (Vendekta, Udekta).
These and other inventions of the 1990s are clearly based on the idea that prefixes would continue backwards through the English or modern Latin alphabet. For example, some on Usenet pointed out that a 1030 prefix (between xenta and vendeka) would have to start with w.
A similar proposal was made by one of the students of H. Paul Shuch, Professor of Electronics at Pennsylvania College of Technology. Shuch wrote a letter to Science News 35 with the student's suggestion of "a linear extrapolation of the recently adopted Zetta, Yotta, zepto, and yocto". Shuch only gives three pairs: Xotta=1027, xocto=10-27, Wotta=1030, wocto=10-30, and Votta=1033, vocto=10-33. His idea was quickly propagated by others including Wikipedia45, and eventually made it into a Saibian article46 extrapolating to a full list of 24 pairs (i.e. all letters a through x) ending with atta=1096 and acto=10-96.
Similarly to Shuch's student, Jim Blowers10 says:
The pattern here is that we go backwards from the beginning of the alphabet [shouldn't this be "end of the alphabet"? -ed], starting with z and y, and we follow it up with an alteration of the Greek or Latin for the next number. According to this pattern, the next ending ["prefix" -ed] should be xona-, since x comes before y in the alphabet, and 9 is noni- in Latin. Similarly, 1030 should be weka-, since w precedes x and 10 is deka in Greek.
He goes on to list a large number of prefixes starting with Xona-, Weka-, Vunda-, Uda-, Treda-, Sorta-, ... One-letter abbreviations are used if unambiguous, otherwise another letter is added, e.g. TD- for the Treda- prefix. He goes as far as 1063 using an L- prefix.
Any of these might be (intentionally or mistakenly) used by others who are sincere, but unaware that the prefix they are using is unofficial, ad-hoc or worse. For example, one reader wrote informing me of the usage of Xera- in a presentation on computer computational speeds by Dr. Laurent Alexandre ("XeraFlops — Un milliard de milliards de milliards opérations par seconde").
Sources
Here is a collection of URLs that are related to the above topics, but which I did not actually use as sources. Most of these are from the beginning of my work on this page (around 1999 or 2000). These pages concern names of large numbers, SI units, and similar things.
(Sources:
Borislav Manolov, Names of large and small numbers, web page.
InterTran English-Latin Translator, via Stars21.
http://www30.brinkster.com/manfear/office/large.html (Also gives names in German and proposes names based on Greek)
http://g42.org/MiscInfo/numbers.html (Much of the same info as on the previous page)
http://www.unc.edu/%7Erowlett/units/large.html (Much of the same info as on the previous page)
http://mathforum.org/library/drmath/view/59155.html (An introduction for kids)
http://www.grammarstation.com/KnowYourMath/numbers_symbols.html (An introduction for kids)
http://www.ex.ac.uk/cimt/dictunit/notesp.htm (Described the SI prefixes for powers of 1024, but now offline)
http://mathworld.wolfram.com/LargeNumber.html
http://journals.iranscience.net:800/www.newscientist.com/www.newscientist.com/lastword/article.jsp@id=lw77 (comments on uses of zetta- and yotta-)
http://www.lewrockwell.com/orig/kinsella6.html (introduction to the SI units)
http://www.plexos.com/256_bit_CPUs_should_be_enough.htm (xona)
http://jimvb.home.mindspring.com/unitsystem.htm (xona)
http://www.omniglot.com/writing/latin.htm (Latin alphabet)
http://www.freezoneearth.org/Prometheus04/otThree/preot3/bignumbers.htm
Wikipedia, List of numbers (Another version that goes up to 10180) )
In the time since the year 2000 there has emerged a dedicated community of people who invent many, many names for specific large numbers. They continue the tradition of the ad-hoc Chuquet system inventors, but take it a lot further with regards to creativity and the sizes of the numbers involved. Their invented names are called googolisms (or googologisms), a suitable name inasmuch as googol is an early example of an arbitrary invented name. (To some people the term "googologism" could refer more generally to any creative effort related to numbers of such size, e.g. a well-defined system of fast-growing functions.)
I call these inventions "ad-hoc googolisms" because they are similar in spirit to the ad-hoc Chuquet names. The similarity is evident in the large number of names invented by each inventor, the use of rules of patterns involving syllables with particular meanings, and definition in terms of the mathematical functions that the particular author uses to handle large numbers. In some cases the inventors seem to wish their "googolisms" to gain widespread usage in the English language, evidenced by the considerable effort they undertake to avoid ambiguities in pronunciation.
Jonathan Bowers
An early and well-known creator is Jonathan Bowers, who had made up a lot of names for examples of each successive fast-growing function in his large number notation system.
myrillion=1030003 micrillion=103000003 (same as Henkle's milli_millillion), killillion=103×103000+3 megillion=103×103000000+3 gigillion=103×103000000000+3 tedakillion=103×103×1042+3 (having run out of SI prefixes) ... giggol = 10④100 (described here) giggolplex = 10④(10④100) trisept = 7(7)7 tridecal = 10(10)10 boogol = 10(100)10 ...
and a ton of names for numbers that are easy to express in his Array Notation, "extended" erray notation, "exploding" arrays, and so on. A lot of these Bowers names appear elsewhere, primarily because Bowers got an early start.
Michael Halm, Razilee Purdue, and André Joyce
Not long after Bowers came the creations attributed to "André Joyce" by "Michael Halm" (which may also be a pseudonym) and translated by "Razilee Purdue" (definitely a pseudonym). Halm's original "Googology" article is part of the "Hierogamous House homepage"; a newer version is this page. on michaelhalm.tripod.com.
The article is entitled simply "GOOGOLOGY", and also called "Mathematics Beyond the Googol" because of the name of the HTML page. The article introduces some novel number names and also summarises a lot of other people's names for special large numbers.
Most of the methods in the article employ word endings, prefixes and infix segments inferred by reverse etymology from existing names like "googol" (10100) and "great gross" (123=1728). The article attributes most of its subject matter to "André Joyce", who is ikely a composite character similar to Martin Gardner's Dr. Matrix used to express the creative output of Halm. According to the article, "Joyce" noticed that googol is "googo" followed by an "l" which he then interpreted as the Roman numeral L (representing 50), and that googol=10100 is equal to 10050, or (2×50)50. He extrapolated this with the generalisation that "googo" followed by any Roman numeral(s) z is (2z)z, which he chose to express as ack-h(2,ack-h(1,2,z),z) using the Herbert version of the Ackermann function. This means that "googoc" would be 200100, "googod" would be 1000500=101500, etc. In another similar generalisation, anything followed by -ple- and Roman numeral(s) such as x is the result of raising x (or whatever) to the power of the thing to which -plex was added (perhaps the quasi-French "André Joyce" imagined "ple" could be an abbreviation of "placé dans l'exposant de"). Thus, for example, twoplex = two, 2, followed by -ple- and x representing 10, is 102; threeplejiv is 33=27 (because "jiv", treated as equivalent to "iiv", is 5 minus 2); and googodplem = 1000googod = 10001000500 = 103×101500.
All the Rest: The Present-Day Googology Community
The Michael Halm40 "GOOGOLOGY" article is most likely the origin of the word googology now better known because of the Googology wiki.
That wiki and related online forums, etc. eventually gained a large membership, and there are thousands of pages for specific ''googolisms'' (invented names). Here are some examples (listed in what I hope is increasing order, but it's tough to be sure):
novopastillion = 10(3×103×1027×1021+3×103×1015×1012+3) named by "Trakaplex" (inventor of over 300 names).
goomolplexiduex = 10↑↑10↑10↑1000 named by "Username5243" (inventor of over 350 names).
forcal = 3↑↑↑↑3, the first step in the construction of the Graham-Gardner number, named by Aarex Tiaokhiao (inventor of over 900 names).
egisitriplextron = M(M(M(M(10,3),3),3),3) in Aarex's "Hyper-Moser notation" (which is equivalent to the Steinhaus-Moser notation), also named by Aarex Tiaokhiao.
gongulusquadraplex = {10,10 (gongulustriplex) 2} in Jonathan Bowers' "Exploding Array" notation, named by Jonathan Bowers (inventor of over 350 names).
truchainpribolplex = s(3,3,3,2,4) in "Hyp cos"'s "strong array notation", named by "Hyp cos" (inventor of almost 600 names).
Empiri-titaniprimol = #{1:1:1}((10))*100 in "SuperJedi224"'s "Hyper-Exploding Pound-Star Notation", named by "SuperJedi224" (inventor of almost 400 names).
godgralgathtothol = E100#^(#^#^#*#^##)100 in Sbiis Saibian41's "Cascading-E notation", named by Sbiis Saibian (inventor of over 6000 names listed on the Googology wiki, plus many thousands more).
bigrand giaquaxul = (...((200![200,200,200,200])![200,200,200,200])![200,200,200,200]...)![200,200,200,200] in Lawrence Hollom's "Hyperfactorial array notation", named by Lawrence Hollom (inventor of over 500 names).
trimixommwil = fψ0(Ωψ0(Ωψ0(Ω)))(10) in the Fast-growing hierarchy, named by Denis Maksudov (inventor of over 600 names).
In nearly all cases the number names come from the person who defined the notation that is used to define the number. This typically happens because the notation systems require extensive explanation with clarifying examples, and the names help make the examples less awkward to present. In most cases, there are patterns used in the names (such as the incorporation of SI prefix syllables), and this makes many of the names only minimally different from each other. Sbiis Saibian41 is notable for having named thousands of numbers, most of them with a single word (perhaps hyphenated).
Sbiis Saibian's Hyper-E (or "E-sharp") notation
This notation has gained some popularity in the Googology community, and is seen in the LNGI incremental game when reaching the final goal. It is called "Hyper-E Notation" by its creator, and abbreviated "E#" by others, and for that reason I sometimes call it "E-sharp".
Its creator Sbiis Saibian41, has written much on the web. A brief description of the original (un-extended) Hyper-E notation is here, (search for the text formal definition for E#). There is one provided example (E303#3 = 101010303, and many more on the subsequent page, many of which are translated into standard notation; and many 'googol'-like invented names.
Ea : Anything with a single E followed by a normal number is just 10 to the power of that number.
EEa or eea : Multiple E's and a single number (but no # signs) are not defined, but many others use this to express a power tower of 10's with the single number a on top: EE100 is googolplex and EEE7 is 1010107. Since this type of abbreviation is described "officially" by Sbiis, it may be better to use lowercase "e", as seen in the LNGI game.
Ea#b : An E followed by two numbers, separated by a single # represents a power-tower of b 10's and a on top. For example, Skewes' number is E34#3. These numbers are not equivalent to more standard notation (such as Conway's up-arrows) except when a is 1. So as a special case, we have:
E1#b : An E followed by 1# and a number b is equal to 10↑↑b = hyper(10,4,b).
Ea#b#2 : An E followed by three numbers the last of which is 2, each separated by #, represents a power-tower of 10's of height Ea#b, with an a on top. For example, following the expansion rules that Saibian gives in his description, starting with E34#3#2 we get E34#(E34#3#1) = E34#(E34#3) = E34#(Skewes), a power-tower of 10's whose height is Skewes' number, with a 34 on top.
It also turns out that (due to the generalised power tower paradox) the general case Ea#b#c is "equivalent" to 10↑↑(c-1) ↑b c using composed up-arrow notation. This is a simplified conversion, and is not exact. Taking the earlier example again, E34#3#2 would be "equivalent" to 10↑↑10↑334 = 10↑↑(10↑(10↑(10↑34))) or 10↑↑Skewes. However, as we showed in the previous paragraph the actual value of E34#3#2 works out to a power tower that has a lot of 10's plus a 34 on the top, whereas 10↑↑Skewes is the same thing without the 34, i.e. a power-tower of all 10's. The missing 34 at the top means that it is more accurate to state:
10↑↑(Skewes+1) < E34#3#2 < 10↑↑(Skewes+2)
Perhaps more usefully, the simplest special case of a 3-argument E-sharp expression converts directly to up-arrows:
E1#1#c : An E followed by 1#1# and a number c is equal to 10↑↑↑c = hyper(10,5,c).
Continuing to more # signs and numbers: for reasons similar to the above, you can add more numbers and # signs and get something that is just a little more, but for most purposes equivalent to, an expression in composed up-arrow notation.
Also, anything with all 1's except the last number is exactly equal to hyper(10,n+2,t) where n is the total number of numeric parts (or one more than the number of # signs) and t is the final numeric part:
E1#1#1#d : An E followed by 1#1#1# and a number d is equal to 10↑↑↑↑d = hyper(10,6,d).
E1#...#1#d : An E followed by n-1 repetitions of "1#" and a number t is equal to 10↑↑...↑↑t (with n up-arrows) which is hyper(10,n+2,t).
E# Extensions
It is soon apparent that one needs to be able to abbreviate runs of the same number. Sbiis Saibian41 provides an "Extended Hyper-E Notation", a "Cascading-E Notation", and more systems beyond that, employing several more symbols and rules to iterate repeated application of anything that can be done with the previously-defined system(s) of symbols and rules.
It starts with ##, which specifies repeated recursive values using a similar string of numbers separated by ## or # but with one number diminished by 1. The substitution is such that it enables values in the range of Graham's number to be reached with three parameters, i.e. Ea##b#c.
In a later extension, the base may also be specified, to get a power-tower of 2 or 3 or some other number. For example "E[3]3##4#64" is an approximation of the Graham-Gardner number.
Huge Numbers in Idle/Clicker Games
There has emerged a class of games, called "Idle" or "Clicker" or "Incremental" games, usually played in a browser. A well-known early example is "Cookie Clicker" by Orteil; another with which I am fairly familiar is Sandcastle Builder by Eternal Density.
These games motivate the player mainly to make some quantity ever larger. Often there is a "score" that continues to grow throughout the game, slowly at first and ever more quickly. Some aspects of human psychology have the effect of making a fixed increase seem smaller when the thing being increased is large. This means that some kind of accelerating growth (such as exponential) is typical of these games. Because of the common property of having gradually increasing values, attained by an endless sequence of small tasks, such games are categorised as "incremental games" (The "Clicker" type place more emphasis on constant interaction, while the "Idle" type place more emphasis on occasional interaction separated by significant gaps during which you can do something else.)
There is a limit to exponential growth in a browser game, and that is usually the IEEE double exponent range, about 10308. To meet the needs of players, an incremental game often bypasses this limit with custom number representations, computational algorithms, and display formats.
The aforementioned Sandcastle Builder is notable for using SI prefixes (including some made-up ones); and Button Simulator Dimension uses abbreviated Conway-Wechsler names; both are discussed in the later section Usage of Words for Large Numbers. All other incremental and idle games just use numbers and special notation like repeated exponent or tetration signs.
Following is a huge list of examples of numbers as they are given by the OmegaNum.js library's string conversion function, and the same numbers as displayed by the game LNGI. Internally it stores the numbers as they would be represented by composed up-arrow notation. LNGI uses "e" to abbreviate "10↑", along with "↑↑" for tetration, and uses E-sharp notation when there is a tetration chain longer than 4 items.
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Gödel's undecidable sentence
The discipline of formal logic was described by Aristotle around 350 BC — although it was probably developed well before him. He described the syllogism, an inference of truth based on other established truths. For example:
if all integers are real numbers
and all primes are integers
then all primes are real numbers
Around 1850, George Boole refined formal logic by defining more precise symbols, axioms and rules, the result is called boolean algebra and propositional calculus.
Meanwhile, set theory was being developed by Georg Cantor and others.
Then Friedrich Ludwig Gottlob Frege began developing the predicate calculus. He applied existing formalism techniques to set theory and began developing a complete formalized system of arithmetic, symbolic logic, and set theory from basic axioms and inference rules. This work was just being published as Grundgesetze der Arithmetik vol. 2 when Bertrand Russell (a mathematician who was also well-known as a writer) presented Frege with the following paradox:
Consider the class of all classes that are not a member of itself. Is this class a member of itself or not?
This paradox challenged Frege so completely that he had to withdraw his work because its foundation was flawed.
After toppling Frege's work, Bertrand Russell and Alfred North Whitehead began work on a similar and even more massive project, with corrections to the problems in Frege's approach and using some of the foundation axioms of Giuseppe Peano. Their goal was to make it possible to derive every true theorem in number theory by starting with a set of axioms and a set of inference rules, and methodically applying all the inference rules to the axioms and existing theorems to create new theorems. For examples, see the book Gödel, Escher Bach [56].
The technique of deriving all truth by an automatic process is appealing — it suggests the possibility of automation (e.g. by a mechanical or electronic computer) to eliminate human error in discovering proofs. However, from the start there was much doubt about whether it could ever be used to discover every truth in number theory (if it were possible, the technique would have completeness). For one thing, there was the issue of time and algorithmic complexity: the number of theorems that are provable in such a system is infinite, and all but a very very tiny fraction of those theorems are completely useless (true, but useless).
The goal of formalization was not questioned, and was continued across various fields of mathematics, e.g. by Hilbert.
Gödel finally showed that the Russell/Whitehead approach would not achieve completeness, in fact that no (sufficiently powerful) axiomatic system of number theory can prove all statements which are true in that system. He did this by showing how statements, expressions and theorems in a number theory system S could be used to encode its own symbols, axioms and inference rules. He then demonstrated that there exists a statement G in that system which asserts "Statement G cannot be derived through the axioms and inference rules of formal system S.". If G is indeed inferrable, then it is false — and the formal system S is inconsistent. But if G cannot be inferred, then it is true, and exists as an example of a true statement that cannot be inferred, and therefore demonstrates that S is incomplete. Thus, all sufficiently powerful formal systems of number theory are either inconsistent or incomplete.
It is now generally agreed that this phenomenon of incompleteness is a property of all axiomatic systems.
Notes on Skewes Number
The Skewes Numbers concern the distribution of primes, specifically how many primes exist that are below a certain number. The Skewes numbers set an upper bound for the point at which the number of primes ends up being less than the approximation "li(x)". See here for a complete description of the problem.
The following table gives the number of primes less than each power of 10:
| n | 10n | primes |
| 1 | 10 | 4 |
| 2 | 100 | 25 |
| 3 | 1,000 | 168 |
| 4 | 10,000 | 1,229 |
| 5 | 100,000 | 9,592 |
| 6 | 1,000,000 | 78,498 |
| 7 | 10,000,000 | 664,579 |
| 8 | 100,000,000 | 5,761,455 |
| 9 | 1,000,000,000 | 50,847,534 |
| 10 | 10,000,000,000 | 455,052,511 |
| 11 | 100,000,000,000 | 4,118,054,813 |
| 12 | 1,000,000,000,000 | 37,607,912,018 |
| 13 | 10,000,000,000,000 | 346,065,536,839 |
| 14 | 100,000,000,000,000 | 3,204,941,750,802 |
| 15 | 1,000,000,000,000,000 | 29,844,570,422,669 |
| 16 | 10,000,000,000,000,000 | 279,238,341,033,925 |
| 17 | 100,000,000,000,000,000 | 2,623,557,157,654,233 |
| 18 | 1,000,000,000,000,000,000 | 24,739,954,287,740,860 |
| 19 | 10,000,000,000,000,000,000 | 234,057,667,276,344,607 |
| 20 | 100,000,000,000,000,000,000 | 2,220,819,602,560,918,840 |
| 21 | 1,000,000,000,000,000,000,000 | 21,127,269,486,018,731,928 |
| 22 | 10,000,000,000,000,000,000,000 | 201,467,286,689,315,906,290 |
(source: Caldwell)
Beyond about n=12 (and slightly larger values as computers get faster) this table cannot be produced by the direct method of finding each prime and counting. There was sufficient interest in this that, as early as 1867, Kulik had completed a table of the smallest factor of every integer (and therefore, giving all the primes) up to slightly above 100,000,000!
Soon after Kulik's table, Meissel found an indirect way to count the exact number of primes up to far higher numbers. His method was simplified by Lehmer in 1959, then improved by Lagarias, Miller and Odlyzko in 1985, and then improved further by Deleglise and Rivat.
The method has been improved to such an extent that it is actually possible to derive values of pi(x) for really huge values of x like 10100 or 101000. In 2005 Patrick Demichel found the actual point at which the pi(x) vs. Li(x) crossover occurs, 1.397162914×10316. .
Comparing the Graham-Gardner Number to Chained Arrow Numbers
The following three numbers are listed in ascending order:
3→3→64→2
the Graham-Gardner number
3→3→65→2.
First, note the function-based definition of the Graham-Gardner number:
gn(n) = { hy(3,6,3) for n = 1 { { hy(3, gn(n-1)+2, 3) for n > 1 Graham-Gardner = gn(64)And we will give another similar definition for a function we'll call c3(N):
c3(n) = { 27 for n = 1 { { hy(3, c3(n-1), 3) for n > 1Here is the expansion process for 3→3→N→2:
3→3→1→2 = 27 = c3(1)
3→3→2→2 = 3→3→(3→3→1→2) = 3→3→c3(1) = hy(3,c3(1),3) = c3(2)
3→3→3→2 = 3→3→(3→3→2→2) = 3→3→c3(2) = hy(3,c3(2),3) = c3(3)
and in general,
3→3→N→2 = c3(N)
Now note that:
c3(1) << gn(1) << c3(2)
c3(2) = hy(3, c3(1), 3) << hy(3, gn(1)+2, 3) = gn(2) << c3(2)
and in general
c3(N) = hy(3, c3(N-1), 3) << hy(3, gn(N-1)+2, 3) = gn(N) << c3(N+1)
therefore,
c3(64) = 3→3→64→2 << gn(64) = Graham-Gardner number << 3→3→65→2 = c3(65)
A reader asked me how 3→3→3→3 compares to these:
3→3→3→3
= 3→3→(3→3→2→3)→2
= 3→3→(3→3→(3→3→1→3)→2)→2
= 3→3→(3→3→27→2)→2
= 3→3→c3(27)→2
= c3(c3(27))
therefore 3→3→3→3 is much bigger than the Graham-Gardner Number.
What Turing and Church Did Not Prove
Turing is often credited with having proven that there are limits to what can be computed by computers, including all conceivable designs of real-world computers. This actually depends a lot on what you mean by computer. Here are some types of computers that a Turing machine cannot emulate:
- Ideal analog computers. This is a computer that manipulates real numbers in the form of analog quantities (like voltages) rather than digits. In an "ideal" analog computer, the quantity being represented is represented exactly, without any error. An example is a circuit that calculates square roots: If you put 2.0 volts on the input wire, it produces 1.4142135... volts on the output.
- Actual analog computers: Any real-world analog computer is subject to various sources of error, including fundamental limits of physics such as the brownian motion of molecules, the quantization of electric charge, and the uncertainty principle (Planck limits).
- Quantum computers: A quantum computer (ideal or actual) exploits properties of quantum mechanics, including most especially the ability for a "particle" such as a photon to be in a state that is indeterminate until that particle interacts with another particle. The conditions of the interaction and the original conditions of the indeterminate state's creation then determine what is observed. It appears that such interactions can be used to perform a sort of massively parallel computation.
- Non-deterministic computers: A nondeterministic computer incorporates some type of phenomenon that is not predictable, in the sense that no amount of advance knowledge can completely predict what the result of a computation will be. Sometimes such phenomena are deliberately exploited, such as a random number generator device based on a Geiger counter (and deriving its randomness from the unpredictable timing of radioactive decay events)
Each of these has one or more properties that make Turing's argument inapplicable. Usually this is some form of unpredictability on a fundamental level. For the ideal analog computers, Turing's argument does not apply because of the cardinality of the reals It is known that Turing planned to create a limits-to-computability proof for such computers, but that he never did (I suspect it was not because such computers are actually without limit!)
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