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12

The quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."

1729

1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):(K. MacMillan, pers. comm., Apr. 29, 2007).This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.1729 was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character..

Strong law of small numbers

The first strong law of small numbers (Gardner 1980, Guy 1988, 1990) states "There aren't enough small numbers to meet the many demands made of them."The second strong law of small numbers (Guy 1990) states that "When two numbers look equal, it ain't necessarily so." Guy (1988) gives 35 examples of this statement, and 40 more in Guy (1990). For example, example 35 notes that the first few values of the interpolating polynomial (erroneously given in Guy (1990) with a coefficient 24 instead of 23) for , 2, ... are 1, 2, 4, 8, 16, .... Thus, the polynomial appears to give the powers of 2, but then continues 31, 57, 99, ... (OEIS A000127). In fact, this sequence gives the maximal number of regions obtained by joining points around a circle by chords (circle division by chords).Similarly, example 41 notes the curious fact that the function where is the ceiling function gives the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (i.e., the first..

239

Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Schroeppel (1972). 239 appears in Machin's formula(1)which is related to the fact that(2)which is why 239/169 is the 7th convergent of . Another pair of inverse tangent formulas involving 239 is(3)(4)239 needs 4 squares (the maximum) to express it, 9 cubes (the maximum, shared only with 23) to express it, and 19 fourth powers (the maximum) to express it (see Waring's problem). However, 239 doesn't need the maximum number of fifth powers (Beeler et al. 1972, Item 63).

Small number

Guy's "strong law of small numbers" states that there aren't enough small numbers to meet the many demands made of them. Guy (1988) also gives several interesting and misleading facts about small numbers: 1. 10% of the first 100 numbers are square numbers.2. A quarter of the numbers are primes. 3. All numbers less than 10, except for 6, are prime powers. 4. Half the numbers less than 10 are Fibonacci numbers.

163

The number 163 is very important in number theory, since is the largest number such that the imaginary quadratic field has class number . It also satisfies the curious identities(1)(2)(3)where is a binomial coefficient.

7

The second Mersenne prime , which is itself the exponent of Mersenne prime . It gives rise to the perfect number It is a Gaussian prime, but not an Eisenstein prime, since it factors as , where is a primitive cube root of unity. It is the smallest non-Sophie Germain prime. It is also the smallest non-Fermat prime, and as such is the smallest number of faces of a regular polygon (the heptagon) that is not constructible by straightedge and compass.It occurs as a sacred number in the Bible and in various other traditions. In Babylonian numerology it was considered as the perfect number, the only number between 2 and 10 which is not generated (divisible) by any other number, nor does it generate (divide) any other number.Words referring to number seven may have the prefix hepta-, derived from the Greek -) (heptic), or sept- (septuple), derived from the Latin septem...

6

The smallest composite squarefree number (), and the third triangular number (). It is the also smallest perfect number, since . The number 6 arises in combinatorics as the binomial coefficient , which appears in Pascal's triangle and counts the 2-subsets of a set with 4 elements. It is also equal to (3 factorial), the number of permutations of three objects, and the order of the symmetric group (which is the smallest non-Abelian group).Six is indicated by the Latin prefix sex-, as in sextic, or by the Greek prefix hexa- (-), as in hexagon, hexagram, or hexahedron.The six-fold symmetry is typical of crystals such as snowflakes. A mathematical and physical treatment can be found in Kepler (Halleux 1975), Descartes (1637), Weyl (1952), and Chandrasekharan (1986).

42

According to the novel The Hitchhiker's Guide to the Galaxy (Adams 1997), 42 is the ultimate answer to life, the universe, and everything. Unfortunately, it is left as an exercise to the reader to determine the actual question.On Feb. 18, 2005, the 42nd Mersenne prime was discovered (Weisstein 2005), leading to humorous speculation that the answer to life, the universe, and everything is somehow contained in the 7.8 million decimal digits of that number.It is also amusing that 042 occurs as the digit string ending at the 50 billionth decimal place in each of and , providing another excellent answer to the ultimate question (Berggren et al. 1997, p. 729).

5

The third prime number, which is also the second Fermat prime, the third Sophie Germain prime, and Fibonacci number . It is an Eisenstein prime, but not a Gaussian prime, since it factors as . It is the hypotenuse of the smallest Pythagorean triple: 3, 4, 5. For the Pythagorean school, the number 5 was the number of marriage, since it is was the sum of the first female number (2) and the first male number (3). The magic symbol of the pentagram was also based on number 5; it is a star polygon with the smallest possible number of sides, and is formed by the diagonals of a regular pentagon. These intersect each other according to the golden ratio .There are five Platonic solids. In algebra, five arises in Abel's impossibility theorem as the smallest degree for which an algebraic equation with general coefficients is not solvable by radicals. According to Galois theory, this property is a consequence of the fact that 5 is the smallest positive integer such that..

4

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of the Pythagorean school, it was the number of justice. The sacred tetraktýs (10) was the sum of the first four numbers, depicted as a triangle with two equal sides of length 4.4 is the highest degree for which an algebraic equation is always solvable by radicals. It is the smallest order of a field which is not a prime field, and the smallest order for which there exist two nonisomorphic finite groups (finite group C2×C2 and the cyclic group C4). It is the smallest number of faces of a regular polyhedron, the tetrahedron. In the three-dimensional Euclidean space, there is exactly one sphere passing through four noncoplanar points. Four is the number of dimensions of space-time.Words related to number four are indicated by the Greek prefix tetra (e.g., tetromino) or by..

3

3 is the only integer which is the sum of the preceding positive integers () and the only number which is the sum of the factorials of the preceding positive integers (). It is also the first odd prime. A quantity taken to the power 3 is said to be cubed.The sequence 1, 31, 331, 3331, 33331, ... (OEIS A033175) consisting of , 1, ... 3s followed by a 1 has its th term is given byThe result is prime for , 2, 3, 4, 5, 6, 7, 17, 39, ... (OEIS A055520); i.e., for 31, 331, 3331, 33331, 333331, 3333331, 33333331, ... (OEIS A051200), a fact which Gardner (1997) calls "a remarkable pattern that is entirely accidental and leads nowhere."

1

The number one (1), also called "unity," is the first positive integer. It is an odd number. Although the number 1 used to be considered a prime number, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own (Wells 1986, p. 31). The number 1 is sometimes also called "unity," so the th roots of 1 are often called the th roots of unity. Fractions having 1 as a numerator are called unit fractions. If only one root, solution, etc., exists to a given problem, the solution is called unique.The generating function having all coefficients1 is given by(1)The number one is also equivalent to the repeatingdecimal(2)(3)

2

The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since . A quantity taken to the power 2 is said to be squared. The number of times a given binary number is divisible by 2 is given by the position of the first , counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.The only known solutions to the congruenceare summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below on Jan. 27 2007, finding no other solutions in this range.reference4700063497Guy (1994)3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)8365386194032363Crump (pers. comm., 2000)10991007971508067Crump (2007)63130707451134435989380140059866138830623361447484274774099906755Montgomery (1999)In general,..

Mega

A large number defined as where the circle notation denotes " in squares," and triangles and squares are expanded in terms of Steinhaus-Moser notation (Steinhaus 1999, pp. 28-29). Here, the typographical error of Steinhaus has been corrected.

24

The number 24 is equal to (four factorial).A number puzzle asks to construct 24 in as many ways possible using elementary mathematical operations on three copies of the same digit. Example solutions include(1)(2)(3)(4)(5)(6)(7)(8)(9)where the not necessarily elementary "operation" of digit concatenation has been indicated using the symbol , any solution with 2 can be replaced with the trivial substitution , and all occurrences of and can be replaced by , , , , etc.

Large number

A wide variety of large numbers crop up in mathematics. Some are contrived, but some actually arise in proofs. Often, it is possible to prove existence theorems by deriving some potentially huge upper limit which is frequently greatly reduced in subsequent versions (e.g., Graham's number, Kolmogorov-Arnold-Moser theorem, Mertens conjecture, Skewes number, Wang's conjecture).Large decimal numbers beginning with are named according to two mutually conflicting nomenclatures: the American system (in which the prefix stands for in ) and the British system (in which the prefix stands for in ). The British names for billion, trillion, etc. originate from the late 15th century when the French physician and mathematician Nicolas Chuquet (1445-1488) used the Latin prefixes to denote successive powers of one million () and the suffix "-llion" to refer one million (Rowlett). In more recent years, the "American" system..

Zillion

A generic word for a very large number. The term has no well-defined mathematical meaning. Conway and Guy (1996) define the th zillion as in the American system (, , , ...) and in the British system (, , , ...). Conway and Guy (1996) also define the words n-plex and n-minex for and , respectively.

Trillion

In the American system, one trillion equals . In the French and German systems, one trillion equals .In recent years, the "American" system has become common in both the United States and Britain. In the words of The Chicago Manual of Style, "The American definitions are gaining acceptance, but writers need to remember the historical geographic distinctions." This use of a common meaning for "trillion" constitutes a fortunate development for standardization of terminology, albeit a somewhat regrettable development from the point of view that the British convention for representing large numbers is simpler and more logical than the American one.

Number

The word "number" is a general term which refers to a member of a given (possibly ordered) set. The meaning of "number" is often clear from context (i.e., does it refer to a complex number, integer, real number, etc.?). Wherever possible in this work, the word "number" is used to refer to quantities which are integers, and "constant" is reserved for nonintegral numbers which have a fixed value. Because terms such as real number, Bernoulli number, and irrational number are commonly used to refer to nonintegral quantities, however, it is not possible to be entirely consistent in nomenclature.To indicate a particular numerical label, the abbreviation "no." is sometimes used (deriving from "numero," the ablative case of the Latin "numerus"), as is the less common "nr." The symbol # (known as the octothorpe) is commonly used to denote "number."While..

Semiprime

A semiprime, also called a 2-almost prime, biprime (Conway et al. 2008), or -number, is a composite number that is the product of two (possibly equal) primes. The first few are 4, 6, 9, 10, 14, 15, 21, 22, ... (OEIS A001358). The first few semiprimes whose factors are distinct (i.e., the squarefree semiprimes) are 6, 10, 14, 15, 21, 22, 26, 33, 34, ... (OEIS A006881).The square of any prime number is by definition a semiprime. The largest known semiprime is therefore the square of the largest known prime.A formula for the number of semiprimes less than or equal to is given by(1)where is the prime counting function and is the th prime (R. G. Wilson V, pers. comm., Feb. 7, 2006; discovered independently by E. Noel and G. Panos around Jan. 2005, pers. comm., Jun. 13, 2006).The numbers of semiprimes less than for , 2, ... are 3, 34, 299, 2625, 23378, 210035, ... (OEIS A066265).For with and distinct, the following..

Billion

In the American system, one billion equals . In the French and German systems, one billion equals .In recent years, the "American" system has become common in both the United States and Britain. In the words of The Chicago Manual of Style (2003), "The American definitions are gaining acceptance, but writers need to remember the historical geographic distinctions." This use of a common meaning for "billion" constitutes a fortunate development for standardization of terminology, albeit a somewhat regrettable development from the point of view that the British convention for representing large numbers is simpler and more logical than the American one.

Jevons' number

A semiprime which English economist and logician William Stanley Jevons incorrectly believed no one else would be able to factor. According to Jevons (1874, p. 123), "Can the reader say what two numbers multiplied together will produce the number 8616460799? I think it unlikely that anyone but myself will ever know."Actually, a modern computer can factor this number in a few milliseconds as the product of two five-digit numbers:Published factorizations include those by Lehmer (1903) and Golomb (1996).

Triskaidekaphobia

Triskaidekaphobia is the fear of 13, a number commonly associated with bad luck in Western culture. While fear of the number 13 can be traced back to medieval times, the word triskaidekaphobia itself is of recent vintage, having been first coined by Coriat (1911; Simpson and Weiner 1992). It seems to have first appeared in the general media in a Nov. 8, 1953 New York Times article covering discussions of a United Nations committee.This superstition leads some people to fear or avoid anything involving the number 13. In particular, this leads to interesting practices such as the numbering of floors as 1, 2, ..., 11, 12, 14, 15, ... (OEIS A011760; the "elevator sequence"), omitting the number 13, in many high-rise American hotels, the numbering of streets avoiding 13th Avenue, and so on.Apparently, 13 hasn't always been considered unlucky. In fact, it appears that in ancient times, 13 was either considered in a positive light or..

10

The number 10 (ten) is the basis for the decimal system of notation. In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number 1234.56 specifies(1)The decimal places to the left of the decimal point are 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, ... (OEIS A011557), called one, ten, hundred, thousand, ten thousand, hundred thousand, million, 10 million, 100 million, and so on. The names of subsequent decimal places for large numbers differ depending on country. Any power of 10 which can be written as the product of two numbers not containing 0s must be of the form for an integer such that neither nor contains any zeros. The largest known such number is(2)A complete list of such known numbers is(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(Madachy 1979). Since..

Milliard

In French and German usage, one milliard equals .American usage does not have a number called the milliard, instead using the term billion to denote . British usage, while formerly using "milliard," has in recent years adopted the American convention (Mish 2003, p. 852). This constitutes a fortunate development for standardization of terminology, albeit a somewhat regrettable development from the point of view that the (former) British convention for representing large numbers is simpler and more logical than the American one.A terrible mathematical joke asks "What American President, with cities in California and Utah named after him, is associated in France and Germany with ?" Answer: Milliard Fillmore (J. vos Post, pers. comm., Apr. 27, 2006).

Beast number

666 is the occult "number of the beast," also called the "sign of the devil" (Wang 1994), associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: "Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666." The origin of this number is not entirely clear, although it may be as simple as the number containing the concatenation of one symbol of each type (excluding ) in Roman numerals: (Wells 1986).The first few numbers containing the beast number in their digits are 666, 1666,2666, 3666, 4666, 5666, 6660, ... (OEIS A051003)."666" is the combination of the mysterious suitcase retrieved by Vincent Vega (John Travolta) and Jules Winnfield (Samuel L. Jackson) in Quentin Tarantino's 1994 film Pulp Fiction. Various conspiracy theories, including the novel..

Sufficiently large

If is a sentential formula depending on a variable ranging in a set of real numbers, the sentence(1)means(2)An example is the proposition(3)which is true, since the inequality is fulfilled for .The statement can also be rephrased as follows: the terms of the sequence become eventually smaller than 0.0001.There are various mathematical jokes involving "sufficiently large." For example, " for sufficiently large values of 1" and "this feature will ship in version 1.0 for sufficiently large values of 1."

Googol

A googol is a large number equal to (i.e., a 1 with 100 zeros following it). Written out explicitly,10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.The term was coined in 1938 by 9-year-old Milton Sirotta, nephew of Edward Kasner (Kasner 1989, pp. 20-27; Bialik 2004). Kasner then extended the term to the larger "googolplex." It should be noted that "googol" is indeed the correct spelling of the term, so the spelling "Google" refers to the internet search engine, not one with 100 zeros.The residues of googol (mod ) for , 2, ... are 0, 0, 1, 0, 0, 4, 4, 0, 1, 0, ... (OEIS A066298).The integer sequence that counts the number of matrices over an alphabet of size 10 and having first few terms 10, 10000, 1000000000, ... (OEIS A076782) reaches googol at its 10th term, and googolplex () at its term. Since grows exponentially, this gives an idea of how..

Zero

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ."The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.Because the number of..

Circle notation

A notation for large numbers due to Steinhaus (1999). In circle notation, is defined as in squares, where numbers written inside squares (and triangles) are interpreted in terms of Steinhaus-Moser notation. The particular number known as the mega is then defined as follows (correcting the typographical error of Steinhaus).

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