1729 is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 the first nontrivial
taxicab number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ''n'' distinct ways. The most famous taxicab numbe ...
, expressed as the
sum of two cubic positive integers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after
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 ...
and
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar
(22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
.
As a natural number
1729 is
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic material ...
, the
squarefree product of three
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s 7 × 13 × 19. It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729. It is the third
Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
, and the first Chernick–Carmichael number. Furthermore, it is the first in the family of absolute
Euler pseudoprimes, a subset of Carmichael numbers. 1729 is divisible by 19, the sum of its digits, making it a
harshad number
In mathematics, a harshad number (or Niven number) in a given radix, number base is an integer that is divisible by the digit sum, sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers ...
in base 10.
1729 is the dimension of the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
on which the fastest known algorithm for multiplying two numbers is based. This is an example of a
galactic algorithm
A galactic algorithm is an algorithm with record-breaking theoretical (asymptotic) performance, but which is not used due to practical constraints. Typical reasons are that the performance gains only appear for problems that are so large they never ...
.
1729 can be expressed as the
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
. Investigating pairs of its distinct integer-valued 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
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
of a four-variable pair is 1729.
Visually, 1729 can be found in other
figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...
s. It is the tenth
centered cube number (a number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points), the nineteenth
dodecagonal number
In mathematics, a dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for ''n'' is given by the formula
:D_=5n^2 - 4n
The first few dodecagonal numbers are:
: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...
(a figurate number in which the arrangement of points resembles the shape of a
dodecagon
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
Regular dodecagon
A regular polygon, regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry ...
), the thirteenth 24-
gonal and the seventh 84-gonal number.
As a Ramanujan number
1729 is also known as ''Ramanujan number'' or ''Hardy–Ramanujan number'', named after an
anecdote
An anecdote is "a story with a point", such as to communicate an abstract idea about a person, place, or thing through the concrete details of a short narrative or to characterize by delineating a specific quirk or trait.
Anecdotes may be real ...
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
Srinivasa Ramanujan Aiyangar
(22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
who was ill in hospital. In their conversation, Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but Ramanujan remarked that "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways". This conversation led to the definition of the
taxicab number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ''n'' distinct ways. The most famous taxicab numbe ...
as the smallest integer that can be expressed as a sum of two positive
cubes
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
in distinct ways. 1729 is the second taxicab number, expressed as
and
.
1729 was later found in one of Ramanujan's notebooks dated years before the incident, and it was noted by French mathematician
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 an affluent district in 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 ...
.
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 number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
, as numbers of the form
, which are also expressible as the sum of two other cubes.
See also
*
1729
Events
January–March
* January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hanover ...
Explanatory footnotes
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 ''Computerphile'' and ''Numberph ...
, 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