Taxicab number

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]

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

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; 19 × 91 = 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 1 + z³ which are also expressible as the sum of two other cubes (sequence A050794 in the OEIS).

Other properties

1729 is a sphenic number. It is the third Carmichael number, and more specifically the first Chernick–Carmichael number (sequence A033502 in the OEIS). Furthermore, it is the first in the family of absolute Euler pseudoprimes, which are a subset of Carmichael numbers.

1729 is 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 ${\displaystyle a^{2}+ab+b^{2}}$ in four different ways with a and b positive integers. The integer pairs ${\displaystyle (a,b)}$ are (25,23), (32,15), (37,8) and (40,3).^[13]

1729 is also the smallest integer side ${\displaystyle d}$ 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.