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1728 (number)

Natural number From Wikipedia, the free encyclopedia

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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.

Quick Facts ← 1727 1728 1729 →, Cardinal ...
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In mathematics

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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]

It is also a Jordan–Pólya number such that it is a product of factorials: .[8][9]

1728 has twenty-eight divisors, which is a perfect count (as with 12, with six divisors). It also has an 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]

.

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".

Modular j-invariant

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 ,[19]

.

Inputting a value of for , where is the imaginary number, yields another cubic integer:

.

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

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.

Other properties

The number of directed open knight's tours in 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]

Decimal digits

Regarding strings of digits of 1728,

  • The sum between 1 and 7 inclusive (as a triangular number) yields 28.
  • Where 1728 is the cube of 12, the sum 1 + 728 = 729 = 93. The digit sum of 1728 is 18.
  • The product of the digits of 1728 is 112, as with 744.
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In culture

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]

See also

References

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