Story of 1729 : The first taxicab number

1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable 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."

G. H. Hardy and Srinivasa Ramanujan 

The two different ways are these:

1729 = 1³ + 12³ = 9³ + 10³

Generalizations of this idea have created the notion of "taxicab numbers" ( The nth taxicab number Ta(n) is the smallest number representable in n ways as a sum of positive cubes).
The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessyin 1657.



Some of the other notable features of 1729 are:

• The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined as numbers of the form 1 + z³ which are also expressible as the sum of two other cubes.

• 1729 is also the third Carmichael number and the first absolute Euler pseudo-prime. It is also a sphenic number.

• 1729 is a Zeisel number. It is a centered cube number, as well as a dodecagonal number, a 24-gonal 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.

• Because in base 10 the number 1729 is divisible by the sum of its digits, it is a Harshad number. It also has this property in octal (1729 = 3301₈, 3 + 3 + 0 + 1 = 7) and hexadecimal(1729 = 6C1₁₆, 6 + C + 1 = 19₁₀), but not in binary.

• 1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first occurrence of all ten digits consecutively in the decimal representation of the transcendental number e.

• Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:

                    1 + 7 + 2 + 9 = 19

                    19 × 91 = 1729.

The physicist Richard Feynman demonstrated his abilities at mental calculation when, during a trip to Brazil, he was challenged to a calculating contest against an experienced abacist. The abacist happened to challenge Feynman to compute the cube root of 1729.03; since Feynman knew that 1729 was equal to 12³+1 (because one cubic foot equals 1728 cubic inches), he was able to compute by hand an accurate value for its cube root using interpolation techniques (specifically, binomial expansion). The abacist had to solve the problem by a more laborious algorithmic method, and lost the competition to Feynman. The anecdote is related by Feynman in his memoir, Surely You're Joking, Mr. Feynman!.