1728 (number)

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross (or grand gross).^[1] It is also the number of cubic inches in a cubic foot.

1728 is the cube of 12,^[2] and therefore equal to the product of the six divisors of 12 (1, 2, 3, 4, 6, 12).^[3] It is also the product of the first four composite numbers (4, 6, 8, and 9), which makes it a compositorial.^[4] As a cubic perfect power,^[5] it is also a highly powerful number that has a record value (18) between the product of the exponents (3 and 6) in its prime factorization.^[6][7]

$${\displaystyle {\begin{aligned}1728&=3^{3}\times 4^{3}=2^{3}\times 6^{3}={\mathbf {12^{3}}}\\1728&=6^{3}+8^{3}+10^{3}\\1728&=24^{2}+24^{2}+24^{2}\\\end{aligned}}}$$

It is also a Jordan–Pólya number such that it is a product of factorials: ${\displaystyle 2!\times (3!)^{2}\times 4!=1728}$.^[8][9]

1728 has twenty-eight divisors, which is a perfect count (as with 12, with six divisors). It also has a Euler totient of 576 or 24², which divides 1728 thrice over.^[10]

1728 is an abundant and semiperfect number, as it is smaller than the sum of its proper divisors yet equal to the sum of a subset of its proper divisors.^[11][12]

It is a practical number as each smaller number is the sum of distinct divisors of 1728,^[13] and an integer-perfect number where its divisors can be partitioned into two disjoint sets with equal sum.^[14]

1728 is 3-smooth, since its only distinct prime factors are 2 and 3.^[15] This also makes 1728 a regular number^[16] which are most useful in the context of powers of 60, the smallest number with twelve divisors:^[17]

${\displaystyle 60^{3}=216000=1728\times 125=12^{3}\times 5^{3}}$.

1728 is also an untouchable number since there is no number whose sum of proper divisors is 1728.^[18]

Many relevant calculations involving 1728 are computed in the duodecimal number system, in-which it is represented as "1000".

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 ${\displaystyle \,{\mathcal {H}}:\{\tau \in \mathbb {C} ,{\text{ }}\mathrm {Im} (\tau )>0\}}$,^[19]

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

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

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

In moonshine theory, the first few terms in the Fourier q-expansion of the normalized j-invariant exapand as,^[20]

${\displaystyle 1728{\text{ }}j(\tau )=1/q+744+196884q+21493760q^{2}+\cdots }$

The Griess algebra (which contains the friendly giant as its automorphism group) and all subsequent graded parts of its infinite-dimensional moonshine module hold dimensional representations whose values are the Fourier coefficients in this q-expansion.

The number of directed open knight's tours in ${\displaystyle 5\times 5}$ minichess is 1728.^[21]

1728 is one less than the first taxicab or Hardy–Ramanujan number 1729, which is the smallest number that can be expressed as sums of two positive cubes in two ways.^[22]

Regarding strings of digits of 1728,

1728 is 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.^[23]

• The year AD 1728

• 1728 at Numbers Aplenty.