Ramanujan Number

1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, 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 and Srinivasa Ramanujan.

As a natural number

1729 is composite, the squarefree product of three prime numbers 7 × 13 × 19. It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729. It is the third Carmichael number, 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 base 10.

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.

1729 can be expressed as the quadratic form. 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 of a four-variable pair is 1729.

Visually, 1729 can be found in other figurate numbers. 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 (a figurate number in which the arrangement of points resembles the shape of a dodecagon), 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 of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan 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 as the smallest integer that can be expressed as a sum of two positive cubes in distinct ways. 1729 is the second taxicab number, expressed as $1^3 + 12^3$ and $9^3 + 10^3$.

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.

The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form $1 + z^3$, which are also expressible as the sum of two other cubes.

See also

* 1729

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, 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?
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Integers
Srinivasa Ramanujan