Английская Википедия:1729 (number)
Шаблон:Short description Шаблон:Infobox number 1729 is the natural number following 1728 and preceding 1730. It is notably the first nontrivial taxicab number.
In mathematics
1729 is the smallest nontrivial taxicab number,[1] and is known as the Hardy–Ramanujan number,[2] after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:[3][4][5][6] Шаблон:Quote
The two different ways are:
- 1729 = 13 + 123 = 93 + 103
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 19Шаблон:Times91 = 1729).
- 91 = 63 + (−5)3 = 43 + 33
1729 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.[7]
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 Шаблон:Math which are also expressible as the sum of two other cubes Шаблон:OEIS.
Other properties
1729 is a sphenic number. It is the third Carmichael number, the first Chernick–Carmichael number Шаблон:OEIS, the first absolute Euler pseudoprime, and the third Zeisel number.[8] It is a centered cube number,[9] as well as a dodecagonal number,[10] a 24-gonal[11] 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.[12]
1729 is the lowest number which can be represented by a Loeschian quadratic form Шаблон:Tmath in four different ways with a and b positive integers. The integer pairs <math>(a,b)</math> are (25,23), (32,15), (37,8) and (40,3).[13]
1729 is also the smallest integer side <math>d</math> of an equilateral triangle for which there are three sets of non-equivalent points at integer distances from their vertices: {211, 1541, 1560}, {195, 1544, 1591}, and {824, 915, 1591}.[14]
1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[15] This is an example of a galactic algorithm.
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
- Шаблон:Mathworld
- Шаблон:Cite web
- Why does the number 1729 show up in so many Futurama episodes?, io9.com
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Quotations by Hardy Шаблон:Webarchive
- ↑ Шаблон:Cite news
- ↑ Шаблон:Cite book
- ↑ Шаблон:Citation The anecdote about 1729 occurs on pages lvii and lviii
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Citation - D1 mentions the Ramanujan-Hardy number.
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
