| Number: | 1729 |
| Divisor: | 1, 7, 13, 19, 91, 133, 247, 1729 |
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is also known as the Ramanujan number or Hardy–Ramanujan number, named after G. H. Hardy and Srinivasa Ramanujan.
1729 is composite, meaning its factors are 1, 7, 13, 19, 91, 133, 247, and 1729. It is the multiplication of its first three smallest prime numbers
7 x 13 x 19
1729 can be defined by summing each of its digits, multiplying by the resulting number with its digit permutably switched, a harshad number. This property can be found in other number systems, such as the octal and hexadecimal. However, this does not work on binary number. It 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.
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 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 otherwise stated it is a number that can be expressed as the sum of two cubic numbers in two different ways. This conversation in the aftermath led to a new class of numbers known as the taxicab number. 1729 is the second taxicab number, expressed as
13+123
93+103
1729 was also found in one of Ramanujan's notebooks dated years before the incident and 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+z3