1728 (number)

1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross).

← 1727 1728 1729 →
List of numbers Integers ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinal	one thousand seven hundred twenty-eight
Ordinal	1728th (one thousand seven hundred twenty-eighth)
Factorization	2^6 × 3^3
Greek numeral	,ΑΨΚΗ´
Roman numeral	MDCCXXVIII
Binary	110110000002
Ternary	21010003
Senary	120006
Octal	33008
Duodecimal	100012
Hexadecimal	6C016

It is the number of cubic inches in a cubic foot.

It is also the number of daily chants of the Hare Krishna mantra by a Hare Krishna devotee. The number comes from 16 rounds on a 108 japamala bead.^[1]

In mathematics

1728 is the cube of 12 and, as such, is important in the duodecimal number system, in which it is represented as "1000".

• 1728 = 12³
• 1728 = 3³ × 4³
• 1728 = 2³ × 6³
• 1728 = 6³ + 8³ + 10³
• 1728 = 24² + 24² + 24²
• 1728 = 289³ + 287³ + (−288)³ + (−288)³
• 28 divisors (perfect count): 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

1728 is the number of directed open knight's tours on a 5 × 5 chessboard.^[2]

1728 occurs in the algebraic formula for the j-invariant of an elliptic curve, as a function over a complex variable on the upper half-plane ${\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}}$ ,^[3]

${\displaystyle j(\tau )=1728{\frac {g_{2}(\tau )^{3}}{\Delta (\tau )}}=1728{\frac {g_{2}(\tau )^{3}}{g_{2}(\tau )^{3}-27g_{3}(\tau )^{2}}}.}$ 

Inputting a value of ${\displaystyle 2i}$  for ${\displaystyle \tau }$ , where ${\displaystyle i}$  is the imaginary number, yields another cubic integer:

${\displaystyle j(2i)=1728{\frac {g_{2}(2i)^{3}}{g_{2}(2i)^{3}-27g_{3}(2i)^{2}}}=66^{3}.}$ 

1728 is one less than the first Hardy–Ramanujan or taxicab number, 1729.^[4]

See Also

• The year 1728 A.D.

References

1. "64 rounds Harināma – Radha Govinda International". Retrieved 2023-03-03.
2. Sloane, N. J. A. (ed.). "Sequence A165134 (Number of directed Hamiltonian paths in the n X n knight graph)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-30.
3. Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340.
4. Sloane, N. J. A. (ed.). "Sequence A011541 (Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-30.