Orders of magnitude (numbers)

Last updated
The logarithmic scale can compactly represent the relationship among variously sized numbers. Logarithmic scale.png
The logarithmic scale can compactly represent the relationship among variously sized numbers.

This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.

Number mathematical object used to count, label, and measure

A number is a mathematical object used to count, measure, and also label. The original examples are the natural numbers 1, 2, 3, 4 and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, number may refer to a symbol, a word, or a mathematical abstraction.

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. Also known as a bare, pure, scalar or a quantity of dimension one.ed and the corresponding unit of measurement in the SI is one unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.


Smaller than 10100 (one googolth)

Monkey animal of the "higher primates" (the simians), but excluding the apes

Monkey is a common name that may refer to groups or species of mammals, in part, the simians of infraorder Simiiformes. The term is applied descriptively to groups of primates, such as families of new world monkeys and old world monkeys. Many monkey species are tree-dwelling (arboreal), although there are species that live primarily on the ground, such as baboons. Most species are also active during the day (diurnal). Monkeys are generally considered to be intelligent, especially the old world monkeys of Catarrhini.

Infinite monkey theorem Counterintuitive result in probability

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would almost surely type every possible finite text an infinite number of times. However, the probability that monkeys filling the observable universe would type a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low.

<i>Hamlet</i> tragedy by William Shakespeare

The Tragedy of Hamlet, Prince of Denmark, often shortened to Hamlet, is a tragedy written by William Shakespeare sometime between 1599 and 1602. Set in Denmark, the play depicts Prince Hamlet and his revenge against his uncle, Claudius, who has murdered Hamlet's father in order to seize his throne and marry Hamlet's mother.

10−100 to 10−30

Standard 52-card deck common card deck used in English-speaking countries

The standard 52-card deck of French playing cards is the most common deck of playing cards used today. It includes thirteen ranks in each of the four French suits: clubs, diamonds, hearts and spades, with reversible "court" or face cards. Each suit includes an ace, a king, queen and jack, each depicted with a symbol of its suit; and ranks two through ten, with each card depicting that many symbols (pips) of its suit. Anywhere from one to six jokers, often distinguishable with one being more colorful than the other, are added to commercial decks, as some card games require these extra cards. Modern playing cards carry index labels on opposite corners or in all four corners to facilitate identifying the cards when they overlap and so that they appear identical for players on opposite sides. The most popular standard pattern of the French deck is sometimes referred to as "English" or "Anglo-American" pattern.

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,


(0.000000000000000000000000000001; 1000−10; short scale: one nonillionth; long scale: one quintillionth)

The long and short scales are two of several large-number naming systems for integer powers of ten that use the same words with different meanings. The long scale is based on powers of one million, whereas the short scale is based on powers of one thousand.

Contract bridge card game

Contract bridge, or simply bridge, is a trick-taking card game using a standard 52-card deck. In its basic format, it is played by four players in two competing partnerships, with partners sitting opposite each other around a table. Millions of people play bridge worldwide in clubs, tournaments, online and with friends at home, making it one of the world's most popular card games, particularly among seniors. The World Bridge Federation (WBF) is the governing body for international competitive bridge, with numerous other bodies governing bridge at the regional level.


(0.000000000000000000000000001; 1000−9; short scale: one octillionth; long scale: one quadrilliardth)


(0.000000000000000000000001; 1000−8; short scale: one septillionth; long scale: one quadrillionth)

ISO: yocto- (y)


(0.000000000000000000001; 1000−7; short scale: one sextillionth; long scale: one trilliardth)

ISO: zepto- (z)


(0.000000000000000001; 1000−6; short scale: one quintillionth; long scale: one trillionth)

ISO: atto- (a)


(0.000000000000001; 1000−5; short scale: one quadrillionth; long scale: one billiardth)

ISO: femto- (f)


(0.000000000001; 1000−4; short scale: one trillionth; long scale: one billionth)

ISO: pico- (p)


(0.000000001; 1000−3; short scale: one billionth; long scale: one milliardth)

ISO: nano- (n)


(0.000001; 1000−2; long and short scales: one millionth)

ISO: micro- (μ)


(0.001; 1000−1; one thousandth)

ISO: milli- (m)


(0.01; one hundredth)

ISO: centi- (c)


(0.1; one tenth)

ISO: deci- (d)


(1; one)


(10; ten)

ISO: deca- (da)


(100; hundred)

ISO: hecto- (h)


(1000; thousand)

ISO: kilo- (k)


(10000; ten thousand or a myriad)


(100000; one hundred thousand or a lakh).


(1000000; 10002; long and short scales: one million)

ISO: mega- (M)


(10000000; a crore; long and short scales: ten million)


(100000000; long and short scales: one hundred million)


(1000000000; 10003; short scale: one billion; long scale: one thousand million, or one milliard)

ISO: giga- (G)


(10000000000; short scale: ten billion; long scale: ten thousand million, or ten milliard)


(100000000000; short scale: one hundred billion; long scale: hundred thousand million, or hundred milliard)


(1000000000000; 10004; short scale: one trillion; long scale: one billion)

ISO: tera- (T)


(1000000000000000; 10005; short scale: one quadrillion; long scale: one thousand billion, or one billiard)

ISO: peta- (P)


(1000000000000000000; 10006; short scale: one quintillion; long scale: one trillion)

ISO: exa- (E)


(1000000000000000000000; 10007; short scale: one sextillion; long scale: one thousand trillion, or one trilliard)

ISO: zetta- (Z)


(1000000000000000000000000; 10008; short scale: one septillion; long scale: one quadrillion)

ISO: yotta- (Y)


(1000000000000000000000000000; 10009; short scale: one octillion; long scale: one thousand quadrillion, or one quadrilliard)


(1000000000000000000000000000000; 100010; short scale: one nonillion; long scale: one quintillion)


(1000000000000000000000000000000000; 100011; short scale: one decillion; long scale: one thousand quintillion, or one quintilliard)


(1000000000000000000000000000000000000; 100012; short scale: one undecillion; long scale: one sextillion)


(1000000000000000000000000000000000000000; 100013; short scale: one duodecillion; long scale: one thousand sextillion, or one sextilliard)

1042 to 10100

(1000000000000000000000000000000000000000000; 100014; short scale: one tredecillion; long scale: one septillion)

10100 (one googol) to 1010100 (one googolplex)

(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; 100033; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard) [47]

Larger than 1010100

(One googolplex; 10googol; short scale: googolplex; long scale: googolplex)

See also

Related Research Articles

<span class="texhtml mvar" style="font-style:italic;">e</span> (mathematical constant) mathematical constant

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

Floating-point arithmetic computer format for representing real numbers

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:

Logarithm A function that maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation (it is an example of a concave function). That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to baseb is denoted as logb (x) (or, without parentheses, as logbx, or even without explicit base as log x, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value.

The number π is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.

Benfords law

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

Rounding replacing numerical value by another approximately equal

Rounding a number means replacing it with a different number that is approximately equal to the original, but has a shorter, simpler, or more explicit representation; for example, replacing $23.4476 with $23.45, or the fraction 312/937 with 1/3, or the expression 2 with 1.414.

Odds are a numerical expression, usually expressed as a pair of numbers, used in both gambling and statistics. In statistics, the odds for or odds of some event reflect the likelihood that the event will take place, while odds against reflect the likelihood that it will not. In gambling, the odds are the ratio of payoff to stake, and do not necessarily reflect exactly the probabilities. Odds are expressed in several ways, and sometimes the term is used incorrectly to mean simply the probability of an event. Conventionally, gambling odds are expressed in the form "X to Y", where X and Y are numbers, and it is implied that the odds are odds against the event on which the gambler is considering wagering. In both gambling and statistics, the 'odds' are a numerical expression of the likelihood of some possible event.

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts.

Power of two two raised to an integer power

In mathematics, a power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent.

In physics and cosmology, digital physics is a collection of theoretical perspectives based on the premise that the universe is describable by information. It is a form of digital ontology about the physical reality. According to this theory, the universe can be conceived of as either the output of a deterministic or probabilistic computer program, a vast, digital computation device, or mathematically isomorphic to such a device.

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently, the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1: the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event. The OR plays an important role in the logistic model, which generalizes beyond two events.

The mathematics of gambling are a collection of probability applications encountered in games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events.

Randomness lack of pattern or predictability in events, or, a measure of uncertainty of an outcome

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.

The hartley, also called a ban, or a dit, is a logarithmic unit which measures information or entropy, based on base 10 logarithms and powers of 10, rather than the powers of 2 and base 2 logarithms which define the bit, or shannon. One ban or hartley is the information content of an event if the probability of that event occurring is ​110. It is therefore equal to the information contained in one decimal digit, assuming a priori equiprobability of each possible value.

A mathematical constant is a special number that is "significantly interesting in some way". Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.


  1. Eric W. Weisstein. "Zero". MathWorld. Retrieved December 10, 2018.
  2. Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 53. ISBN   978-0-7167-1088-2.
  3. There are around 130,000 letters and 199,749 total characters in Hamlet; 26 letters ×2 for capitalization, 12 for punctuation characters = 64, 64199749 10360,783.
  4. Robert Matthews. "What are the odds of shuffling a deck of cards into the right order?". Science Focus. Retrieved December 10, 2018.
  5. www.BridgeHands.com, Sales. "Probabilities Miscellaneous: Bridge Odds". Archived from the original on 2009-10-03.
  6. Walraven, P. L.; Lebeek, H. J. (1963). "Foveal Sensitivity of the Human Eye in the Near Infrared". J. Opt. Soc. Am. 53 (6): 765–766. doi:10.1364/josa.53.000765.
  7. Courtney Taylor. "The Probability of Being Dealt a Royal Flush in Poker". ThoughtCo. Retrieved December 10, 2018.
  8. Mason, W S; Seal, G; Summers, J (1980-12-01). "Virus of Pekin ducks with structural and biological relatedness to human hepatitis B virus". Journal of Virology. 36 (3): 829–836. ISSN   0022-538X. PMC   353710 . PMID   7463557.
  9. 1 2 "Homo sapiens – Ensembl genome browser 87". www.ensembl.org. Archived from the original on 2017-05-25. Retrieved 2017-01-28.
  10. "Pi World Ranking List". Archived from the original on 2017-06-29.
  11. Sloane, N. J. A. (ed.). "SequenceA283670". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2017-03-15.
  12. Plouffe's Inverter Archived 2005-08-12 at the Wayback Machine
  13. Christof Baron (2015). "Facebook users worldwide 2016 | Statista". Statista. statista.com. Archived from the original on 2016-09-09.
  14. 1 2 "Earth microbes on the moon". Science@Nasa. 1 September 1998. Archived from the original on 23 March 2010. Retrieved 2 November 2010.
  15. "there was, to our knowledge, no actual, direct estimate of numbers of cells or of neurons in the entire human brain to be cited until 2009. A reasonable approximation was provided by Williams and Herrup (1988), from the compilation of partial numbers in the literature. These authors estimated the number of neurons in the human brain at about 85 billion [...] With more recent estimates of 21–26 billion neurons in the cerebral cortex (Pelvig et al., 2008 ) and 101 billion neurons in the cerebellum (Andersen et al., 1992 ), however, the total number of neurons in the human brain would increase to over 120 billion neurons." Herculano-Houzel, Suzana (2009). "The human brain in numbers: a linearly scaled-up primate brain". Front. Hum. Neurosci. 3: 31. doi:10.3389/neuro.09.031.2009. PMC   2776484 . PMID   19915731.
  16. Kapitsa, Sergei P (1996). "The phenomenological theory of world population growth". Physics-Uspekhi. 39 (1): 57–71. Bibcode:1996PhyU...39...57K. doi:10.1070/pu1996v039n01abeh000127. (citing the range of 80 to 150 billion, citing K. M. Weiss, Human Biology 56637, 1984, and N. Keyfitz, Applied Mathematical Demography, New York: Wiley, 1977). C. Haub, "How Many People Have Ever Lived on Earth?", Population Today 23.2), pp. 5–6, cited an estimate of 105 billion births since 50,000 BC, updated to 107 billion as of 2011 in Haub, Carl (October 2011). "How Many People Have Ever Lived on Earth?". Population Reference Bureau. Archived from the original on April 24, 2013. Retrieved April 29, 2013. (due to the high infant mortality in pre-modern times, close to half of this number would not have lived past infancy).
  17. Elizabeth Howell, How Many Stars Are in the Milky Way? Archived 2016-05-28 at the Wayback Machine , Space.com, 21 May 2014 (citing estimates from 100 to 400 billion).
  18. Hollis, Morgan (13 October 2016). "A universe of two trillion galaxies". The Royal Astronomical Society. Retrieved 9 November 2017.
  19. Jonathan Amos (3 September 2015). "Earth's trees number 'three trillion'". BBC. Archived from the original on 6 June 2017.
  20. Xavier Gourdon (October 2004). "Computation of zeros of the Zeta function". Archived from the original on 15 January 2011. Retrieved 2 November 2010.
  21. Alexander J. Yee & Shigeru Kondo (28 Dec 2013). "12.1 Trillion Digits of Pi". Archived from the original on 2014-02-21. Retrieved 17 Feb 2014.
  22. Koch, Christof. Biophysics of computation: information processing in single neurons. Oxford university press, 2004.
  23. Savage, D. C. (1977). "Microbial Ecology of the Gastrointestinal Tract". Annual Review of Microbiology. 31: 107–33. doi:10.1146/annurev.mi.31.100177.000543. PMID   334036.
  24. Berg, R. (1996). "The indigenous gastrointestinal microflora". Trends in Microbiology. 4 (11): 430–5. doi:10.1016/0966-842X(96)10057-3. PMID   8950812.
  25. Bert Holldobler and E.O. Wilson The Superorganism: The Beauty, Elegance, and Strangeness of Insect Societies New York:2009 W.W. Norton Page 5
  26. "60th Birthday of Microelectronics Industry". Semiconductor Industry Association. 13 December 2007. Archived from the original on 13 October 2008. Retrieved 2 November 2010.
  27. Sequence A131646 Archived 2011-09-01 at the Wayback Machine in The On-Line Encyclopedia of Integer Sequences
  28. "Smithsonian Encyclopedia: Number of Insects Archived 2016-12-28 at the Wayback Machine ". Prepared by the Department of Systematic Biology, Entomology Section, National Museum of Natural History, in cooperation with Public Inquiry Services, Smithsonian Institution. Accessed 27 December 2016. Facts about numbers of insects. Puts the number of individual insects on Earth at about 10 quintillion (1019).
  29. Ivan Moscovich, 1000 playthinks: puzzles, paradoxes, illusions & games, Workman Pub., 2001 ISBN   0-7611-1826-8 .
  30. "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. Archived from the original on 1 March 2009. Retrieved 14 March 2009.
  31. "To see the Universe in a Grain of Taranaki Sand". Archived from the original on 2012-06-30.
  32. "Intel predicts 1,200 quintillion transistors in the world by 2015". Archived from the original on 2013-04-05.
  33. "How Many Transistors Have Ever Shipped? – Forbes". Archived from the original on 30 June 2015. Retrieved 1 September 2015.
  34. "Sudoku enumeration". Archived from the original on 2006-10-06.
  35. "Star count: ANU astronomer makes best yet". The Australian National University. 17 July 2003. Archived from the original on July 24, 2005. Retrieved 2 November 2010.
  36. "Astronomers count the stars". BBC News. July 22, 2003. Archived from the original on August 13, 2006. Retrieved 2006-07-18. "trillions-of-earths-could-be-orbiting-300-sextillion-stars" van Dokkum, Pieter G.; Charlie Conroy (2010). "A substantial population of low-mass stars in luminous elliptical galaxies". Nature. 468 (7326): 940–942. arXiv: 1009.5992 . Bibcode:2010Natur.468..940V. doi:10.1038/nature09578. PMID   21124316. "How many stars?" Archived 2013-01-22 at the Wayback Machine ; see mass of the observable universe
  37. (sequence A007377 in the OEIS )
  38. "Questions and Answers – How many atoms are in the human body?". Archived from the original on 2003-10-06.
  39. William B. Whitman; David C. Coleman; William J. Wiebe (1998). "Prokaryotes: The unseen majority". Proceedings of the National Academy of Sciences of the United States of America. 95 (12): 6578–6583. Bibcode:1998PNAS...95.6578W. doi:10.1073/pnas.95.12.6578. PMC   33863 . PMID   9618454.
  40. (sequence A070177 in the OEIS )
  41. (sequence A035064 in the OEIS )
  42. John Tromp (2010). "John's Chess Playground". Archived from the original on 2014-06-01.
  43. Planck Collaboration (2016). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". Astronomy & Astrophysics. 594: A13. arXiv: 1502.01589 . Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830.
  44. Paul Zimmermann, "50 largest factors found by ECM Archived 2009-02-20 at the Wayback Machine ".
  45. Matthew Champion, "Re: How many atoms make up the universe?" Archived 2012-05-11 at the Wayback Machine , 1998
  46. WMAP- Content of the Universe Archived 2016-07-26 at the Wayback Machine . Map.gsfc.nasa.gov (2010-04-16). Retrieved on 2011-05-01.
  47. "Names of large and small numbers". bmanolov.free.fr. Miscellaneous pages by Borislav Manolov. Archived from the original on 2016-09-30.
  48. "Richard Eldridge".
  49. Chris Caldwell, The Top Twenty: Elliptic Curve Primality Proof at The Prime Pages.
  50. Chris Caldwell, The Top Twenty: Twin Primes Archived 2013-01-27 at the Wayback Machine at The Prime Pages.
  51. Chris Caldwell, The Top Twenty: Sophie Germain (p) at The Prime Pages.
  52. Chris Caldwell, The Top Twenty: Palindrome at The Prime Pages.
  53. PrimeGrid's Primorial Prime Search Archived 2013-11-26 at the Wayback Machine
  54. Chris Caldwell, The Top Twenty: Factorial primes Archived 2013-04-10 at the Wayback Machine at The Prime Pages.
  55. From the third paragraph of the story: "Each book contains 410 pages; each page, 40 lines; each line, about 80 black letters." That makes 410 x 40 x 80 = 1,312,000 characters. The fifth paragraph tells us that "there are 25 orthographic symbols" including spaces and punctuation. The magnitude of the resulting number is found by taking logarithms. However, this calculation only gives a lower bound on the number of books as it does not take into account variations in the titles – the narrator does not specify a limit on the number of characters on the spine. For further discussion of this, see Bloch, William Goldbloom. The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press: Oxford, 2008.
  56. Chris Caldwell, The Top Twenty: Generalized Fermat Archived 2014-12-23 at the Wayback Machine at The Prime Pages.
  57. Chris Caldwell, The Top Twenty: Proth at The Prime Pages.
  58. 1 2 Chris Caldwell, The Top Twenty: Largest Known Primes at The Prime Pages.
  59. Chris Caldwell, Mersenne Primes: History, Theorems and Lists at The Prime Pages.
  60. asantos (15 December 2007). "Googol and Googolplex by Carl Sagan" via YouTube.
  61. Zyga, Lisa "Physicists Calculate Number of Parallel Universes" Archived 2011-06-06 at the Wayback Machine , PhysOrg , 16 October 2009.