Hardy - Ramanujan number

1729 is refered as Hardy - Ramanujan number. There is an interesting story behind this. The history and pecularity of the number is described below.

History of Hardy - Ramanujan number 1729

Once, in a taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

1729=1³+12³=9³+10³

The smallest number of this type, that can be expressed as the sum of two cubes in n distinct ways are sometimes called Taxicab Numbers.

The Hardy- Ramanujan number that 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

Other pecularities of the number 1729

• First in the sequence of "Fermat near misses” (numbers of the form 1 + z3 which are also expressible as the sum of two other cubes).

• Third Carmichael number.
• The first absolute Euler pseudoprime.
• It is also a sphenic number.
• 1729 is a Zeisel number.
•  It is 10th centered cube number.
•  It is a dodecagonal number, a 24-gonal and 84-gonal number.
• It is a Harshad number (in base 10 the number 1729 is divisible by the sum of its digits). 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.

• 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

Definitions of terms mentioned above

Carmichael number

A composite positive integer n which satisfies the congruence

b^n-1 = 1 (mod n)

for all integers b which are relatively prime to n.

Euler pseudoprime

An odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and

a^(n-1)/2 = 1 (mod n)

(where mod refers to the modulo operation).

Sphenic number

 Sphenic number is a positive integer which is the product of three distinct prime numbers.

Zeisel number

Named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern

p_x = ap_{x − 1} + b

where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest.

Zeisel number starts from 105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979 etc.

Centered cube number

A centered cube number is a centered figurate number that represents a cube. The centered cube number for n is given by

n3 + (n + 1)3.

Centered cube numbers till 1729

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729