1729 is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
following
1728
Events
January–March
* January 5 – The '' Real y Pontificia Universidad de San Gerónimo de la Habana'', the oldest university in Cuba, is founded in Havana.
* January 9 – The coronation of Peter II as the Tsar of t ...
and preceding 1730. It is a
taxicab number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), also called the ''n''th Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ...
, and is variously known as Ramanujan's number and the Ramanujan-Hardy number, after an anecdote of the British mathematician
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
when he visited Indian mathematician
Srinivasa Ramanujan in hospital. He related their conversation:
The two different ways are:
: 1729 = 1
3 + 12
3 = 9
3 + 10
3
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a
negative integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
) gives the smallest solution as
91 (which is a
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of 1729; 1991 = 1729).
:91 = 6
3 + (−5)
3 = 4
3 + 3
3
Numbers that are the smallest number that can be expressed as the sum of two cubes in ''n'' distinct ways have been dubbed "
taxicab number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), also called the ''n''th Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ...
s". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by
Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy incident, at 2 Colinette Road in
Putney
Putney () is a district of southwest London, England, in the London Borough of Wandsworth, southwest of Charing Cross. The area is identified in the London Plan as one of 35 major centres in Greater London.
History
Putney is an ancient paris ...
.
The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
, as numbers of the form which are also expressible as the sum of two other cubes.
Other properties
1729 is also the third
Carmichael number
In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation:
:b^n\equiv b\pmod
for all integers b. The relation may also be expressed in the form:
:b^\equiv 1\pmod.
for all integers ...
, the first Chernick–Carmichael number , and the first absolute
Euler pseudoprime In arithmetic, an odd composite integer ''n'' is called an Euler pseudoprime to base ''a'', if ''a'' and ''n'' are coprime, and
: a^ \equiv \pm 1\pmod
(where ''mod'' refers to the modulo operation).
The motivation for this definition is the f ...
. It is also a
sphenic number
In number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.
Definit ...
.
1729 is also the third
Zeisel number A Zeisel number, named after Helmut Zeisel, is a square-free integer ''k'' with at least three prime factors which fall into the pattern
:p_x = ap_ + b
where ''a'' and ''b'' are some integer constants and ''x'' is the index number of each prime f ...
. It is a
centered cube number
A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with points on the square faces of the th layer. Equival ...
, as well as a
dodecagonal number, a 24-
gonal and 84-gonal number.
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.
1729 is the lowest number which can be represented by a
Loeschian quadratic form in four different ways with ''a'' and ''b'' positive integers. The integer pairs (''a'',''b'') are (25,23), (32,15), (37,8) and (40,3).
1729 is the dimension of the
Fourier transform on which the fastest known algorithm for
multiplying two numbers is based.
This is an example of a
galactic algorithm
A galactic algorithm is one that outperforms any other algorithm for problems that are sufficiently large, but where "sufficiently large" is so big that the algorithm is never used in practice. Galactic algorithms were so named by Richard Lipton ...
.
See also
* ''
A Disappearing Number'', a March 2007 play about Ramanujan in England during World War I.
*
Interesting number paradox
*
4104, the second positive integer which can be expressed as the sum of two positive cubes in two different ways.
References
External links
*
* {{cite web, last=Grime, first=James, title=1729: Taxi Cab Number or Hardy-Ramanujan Number, url=http://www.numberphile.com/videos/1729taxicab.html, work=Numberphile, publisher=
Brady Haran
Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Periodic Videos'' and '' Numbe ...
, author2=Bowley, Roger, access-date=2013-04-02, archive-url=https://web.archive.org/web/20170306141337/http://numberphile.com/videos/1729taxicab.html, archive-date=2017-03-06, url-status=dead
Why does the number 1729 show up in so many Futurama episodes? io9.com
Integers
Srinivasa Ramanujan