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Why 1729 number is so special?

1729 is the natural number following 1728 and preceding 1730. It is known as 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:


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 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):91 = 63 + (−5)3 = 43 + 33

Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways[5] 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.

The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in the OEIS) 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.

1729 is also the third Carmichael number and the first absolute Euler pseudoprime. It is also a sphenic number.

1729 is a Zeisel number.[6] It is a centered cube number,[7] as well as a dodecagonal number,[8] a 24-gonal[9] 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 (Guy 2004).

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 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 (a,b) are (25,23), (32,15), (37,8) and (40,3).[10]

1729 has another mildly interesting property: the 1729th decimal place is the beginning of the first consecutive occurrence of all ten digits without repetition in the decimal representation of the transcendental number e.[11]

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
It suffices only to check sums congruent to 0 or 1 (mod 9) up to 19.

A taxicab number is the name given by mathematicians to a series of special numbers: 2, 1729 etc. A taxicab number is the smallest number that can be expressed as the sum of two positive cubes in n distinct ways. It has nothing to do with taxis, but the name comes from a well-known conversation that took place between two famous mathematicians: Godfrey Hardy and Srinivasa Ramanujan.

← 172817291730 →

 0 1k 2k 3k 4k 5k 6k 7k 8k 9k 
Cardinalone thousand seven hundred twenty-nine
Ordinal1729th
(one thousand seven hundred twenty-ninth)
Factorization7 × 13 × 19
Divisors1, 7, 13, 19, 91, 133, 247, 1729
Greek numeral,ΑΨΚΘ´
Roman numeralMDCCXXIX
Greek prefix2
Latin prefix1
Binary110110000012
Ternary21010013
Quaternary1230014
Quinary234045
Senary120016
Octal33018
Duodecimal100112
Hexadecimal6C116
Vigesimal46920
Base 361C136

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