# 1728 (number)

(Redirected from Great gross)
 ← 1727 1728 1729 →
Cardinalone thousand seven hundred twenty-eight
Ordinal1728th
(one thousand seven hundred twenty-eighth)
Factorization26 × 33
Greek numeral,ΑΨΚΗ´
Roman numeralMDCCXXVIII
Binary110110000002
Ternary21010003
Senary120006
Octal33008
Duodecimal100012

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

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]