List of numbers — Integers << 1k 2k 3k 4k 5k 6k 7k 8k 9k >> This is a list of articles about numbers (not about numerals). ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Nine hundred is the natural number following eight hundred ninety-nine and preceding nine hundred one. ... Cardinal 1000 one thousand Ordinal 1000th Numeral system Factorization Prime Divisor(s) Roman numeral Unicode symbol(s) , , Greek Prefix chilia Latin Prefix milli Binary 1111101000 Octal 1750 Duodecimal 6B4 Hexadecimal 3E8 1000 (one thousand) is the natural number following 999 and preceding 1001. ... 2000 (two thousand) is the natural number following 1999 and preceding 2001. ... Three thousand (3000) is the natural number following 2999 and preceding 3001. ... Four thousand (4000) is the natural number following 3999 and preceding 4001. ... Five thousand (5000) is the natural number following 4999 and preceding 5001. ... Six thousand (6000) is the natural number following 5999 and preceding 6001. ... 7000 is the natural number following 6999 and preceding 7001. ... 8000 (eight thousand) is the natural number following 7999 and preceding 8001. ... 9000 is the natural number following 8999 and preceding 9001. ... Ten thousand (10,000) is the natural number following 9999 and preceding 10,001. ...
1729
Cardinal	One thousand seven hundred [and] twenty-nine
Ordinal	1729th
Factorization
Divisors	7, 13, 19, 91, 133, 247
Roman numeral	MDCCXXIX
Binary	11011000001
Octal	3301
Duodecimal	1001
Hexadecimal	6C1

1729 is known as the Hardy-Ramanujan number, after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words [1]: In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... The octal numeral system is the base-8 number system, and uses the digits 0 to 7. ... A duodecimal multiplication table The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ... Srinivasa Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician and one of the greatest mathematical geniuses of the twentieth century. ...

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

The quote is sometimes expressed using the term "positive cubes", as the admission of negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a factor of 1729): Image File history File links Cquote1. ... Putney is a middle-class district in the London Borough of Wandsworth. ... now. ... Image File history File links Cquote2. ... A negative number is a number that is less than zero, such as −3. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... 91 (ninety-one) is the natural number following 90 and preceding 92. ...

91 = 6³ + (−5)³ = 4³ + 3³

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like −91, −189, −1729, and further negative numbers. This ambiguity is eliminated by the term "positive cubes".

Numbers such as

1729 = 1³ + 12³ = 9³ + 10³

that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed taxicab numbers. 1729 is the second taxicab number (the first is 2 = 1³ + 1³). The number was also found in one of Ramanujan's notebooks dated years before the incident. In mathematics, the n-th taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number which can be expressed as a sum of two positive cubes in n distinct ways, up to order of summands. ...

1729 is the third Carmichael number, and a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal and 84-gonal number. In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn âˆ’ 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ... A Zeisel number is a square-free integer k with at least three prime factors which fall into the pattern where a and b are fixed constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest. ... A centered cube number is a centered figurate number that represents a cube. ... A dodecagonal number is a figurate number that represents a dodecagon. ... In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...

Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729 (Guy 2004). In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, a discriminant is an expression which discriminates qualities of algebraic structures. ...

Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal (1729 = 3301₈, 3 + 3 + 0 + 1 = 7) and hexadecimal (1729 = 6C1₁₆, 6 + C + 1 = 19₁₀), but not in binary. A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base. ... The octal numeral system is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...

1729 has another interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits without repetition in the decimal representation of the transcendental number e, although, of course, this fact would not have been known to either mathematician, since the computer algorithms used to discover this were not implemented until much later. [2] In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ...

Masahiko Fujiwara showed that 1729 is one of four natural numbers (the others are 81 and 1458 and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversed self, yields the original number: Masahiko Fujiwara is a Japanese mathematician who is best known for his essays. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... 81 is the natural number following 80 and preceding 82. ... 1458 Cardinal One thousand four hundred [and] fifty-eight Ordinal 1458th Factorization Divisors 3,6,9,18,21,42,63,126,189,378,729 Roman numeral MCDLVIII Binary 10110110010 Hexadecimal 5B2 1458 is one of three numbers (the other two are 81 and 1729) which, when its digits are added... Look up one in Wiktionary, the free dictionary. ...

1 + 7 + 2 + 9 = 19

19 · 91 = 1729

It has occasionally been suggested that Hardy's story is apocryphal, on the grounds that he almost certainly would have been familiar with some of these features of the number.^{[citation needed]}

References to 1729

The television show Futurama contains several jokes about the Hardy-Ramanujan number. In one episode, the robot Bender receives a Christmas card from the machine that built him labeled "Son #1729". Ken Keeler, a writer on the show with a Ph. D. in Applied Math, said that "that 'joke' alone is worth six years of grad school." In another episode, Bender's serial number is revealed to be the sum of two cubes: his number is 2716057 = 952³ + (−951)³, while that of fellow robot Flexo is 3370318 = 119³ + 119³. (This datum is one of the pieces of evidence the episode uses to establish that Bender and Flexo are a pair of good-and-evil twins.) The starship Nimbus displays the hull registry number BP-1729, which simultaneously riffs on the USS Enterprise's NCC-1701. Finally, the episode The Farnsworth Parabox contains a montage sequence where the heroes visit several parallel universes in rapid succession, one of which is labeled "Universe 1729". Futurama is an animated American cartoon series created by Matt Groening (creator of The Simpsons) and David X. Cohen (also a writer for The Simpsons). ... A robot is an electro-mechanical device that can perform autonomous or preprogrammed tasks. ... Bender Bending Rodríguez, more commonly known as Bender (assembled c. ... Keeler at the 2003 Writers Guild Awards, after winning in the animation category. ... USS Enterprise (XCV 330) Enterprise (NX-01) (from United Earth’s Starfleet) USS Enterprise (NCC-1701) (the Federations first so named) USS Enterprise (NCC-1701-A) (the Federations second so named) USS Enterprise (NCC-1701-B) (the Federations third so named) USS Enterprise (NCC-1701-C) (the... The Farnsworth Parabox is an episode of Futurama Plot Summary Template:Spoler After nealry dying because of an experiment, Farnsworth orders the crew to never open his experiment; a yellow box. ...

The physicist Richard Feynman demonstrated his abilities at mental calculation when, during a trip to Brazil, he was challenged to a calculating contest against an experienced abacist. The abacist happened to challenge Feynman to compute the cube root of 1729.03; since Feynman knew that 1729 was equal to 12³+1, he was able to give an accurate value for its cube root mentally using interpolation techniques (specifically, binomial expansion). The abacist had to solve the problem by a more laborious algorithmic method, and lost the competition to Feynman. What I cannot create, I do not understand —Richard P. Feynman Richard Phillips Feynman (May 11, 1918 in Queens, New York – February 15, 1988 in Los Angeles, California) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, particle... Mental calculation is the practice of doing mathematical calculations using only the human brain, with no help from any computing devices. ... An abacus is a calculation tool, often constructed as a wooden frame with beads sliding on wires. ... Plot of y = In mathematics, the cube root ( ) of a number is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

Some reports say that the octal equivalent (3301) was the password to Xerox PARC's main computer. Bold text // Headline text Link title This article is about the computer research center. ...

The play Proof (and its adapted film) by David Auburn also contains a reference to 1729. Proof is a play by David Auburn which won the 2001 Pulitzer Prize for Drama and the 2001 Tony Award for Best Play. ... David Auburn (born 1969) is an American playwright. ...

The movie Lucky Number Slevin also references the number 1729 in association with the character Nick Fisher. Lucky Number Slevin is a 2006 gangster film written by Jason Smilovic, directed by Paul McGuigan and starring Bruce Willis, Josh Hartnett, Lucy Liu, Morgan Freeman and Ben Kingsley. ...

Quotation

• "Every positive integer is one of Ramanujan's personal friends."—J. E. Littlewood, on hearing of the taxicab incident.

John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ...

See also

• Interesting number paradox
• Berry paradox

The interesting number paradox is a semi-humorous paradox that arises from attempting to classify numbers as interesting or dull. ... The Berry paradox is the apparent contradiction that arises from expressions such as the following: The smallest positive integer not nameable in under eleven words. ...

References

• Martin Gardner, Mathematical Puzzles and Diversions, 1959
• Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer, 2004. D1 mentions the Hardy-Ramanujan number.

Martin Gardner (b. ... Richard Kenneth Guy (born 1916) is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ...

External links

• MathWorld: Hardy-Ramanujan Number
• The Dullness of 1729