22/12/2018
Why is 1729 known as Ramanujan's number?
1729 is also the third 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 (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.
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.[5]
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.
