1729 (number)


1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Hardy–Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:
The two different ways are:
The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes gives the smallest solution as 91 :
Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed "taxicab numbers". The number 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 Hardy–Ramanujan 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 which are also expressible as the sum of two other cubes.

Other properties

1729 is also the third Carmichael number, the first Chernick–Carmichael number, and the first absolute Euler pseudoprime. 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 and hexadecimal, but not in binary and duodecimal.
In base 12, 1729 is written as 1001, so its reciprocal has only period 6 in that base.
1729 is the lowest number which can be represented by a Loeschian quadratic form a² + ab + b² in four different ways with a and b positive integers. The integer pairs are,, and.
1729 has another mildly interesting property: the 1729th digit is the beginning of the first consecutive occurrence of all ten digits without repetition in the decimal representation of the transcendental number e.
Masahiko Fujiwara showed that 1729 is one of four positive integers which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:
It suffices only to check sums congruent to 0 or 1 up to 19.