1728 (number)

From HandWiki
Short description: Natural number
← 1727 1728 1729 →
Cardinalone thousand seven hundred twenty-eight
Ordinal1728th
(one thousand seven hundred twenty-eighth)
Factorization26 × 33
Divisors1, 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
Greek numeral,ΑΨΚΗ´
Roman numeralMDCCXXVIII
Binary110110000002
Ternary21010003
Quaternary1230004
Quinary234035
Senary120006
Octal33008
Duodecimal100012
Hexadecimal6C016
Vigesimal46820
Base 361C036

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.

In mathematics

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]

[math]\displaystyle{ \begin{align} 1728& = 3^{3} \times4^{3} = 2^{3} \times 6^{3} = \bold {12^{3}} \\ 1728& = 6^{3} + 8^{3} + 10^{3} \\ 1728& = 24^{2} + 24^{2} + 24^{2} \\ \end{align} }[/math]

It is also a Jordan–Pólya number such that it is a product of factorials: [math]\displaystyle{ 2! \times (3!)^{2} \times4! = 1728. }[/math][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]

[math]\displaystyle{ 60^{3} = 216000 = 1728 \times 125 = 12^{3} \times 5^{3} }[/math]

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 [math]\displaystyle{ \,\mathcal {H}: \{\tau \in \mathbb {C}, \text{ }\mathrm{Im}(\tau)\gt 0\} }[/math],[19]

[math]\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}. }[/math]

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

[math]\displaystyle{ j(2i) = 1728 \frac{g_2(2i)^3}{g_2(2i)^3 - 27g_3(2i)^2} = 66^3. }[/math]

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

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

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 [math]\displaystyle{ 5 \times 5 }[/math] 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]

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

  • The year AD 1728

References

  1. ↑ "Great gross (noun)". Merriam-Webster, Inc.. https://www.merriam-webster.com/dictionary/great%20gross. 
  2. ↑ Sloane, N. J. A., ed. "Sequence A000578 (The cubes.)". OEIS Foundation. https://oeis.org/A000578. Retrieved 2023-04-03. 
  3. ↑ Sloane, N. J. A., ed. "Sequence A007955 (Product of divisors of n.)". OEIS Foundation. https://oeis.org/A007955. Retrieved 2023-04-03. 
  4. ↑ Sloane, N. J. A., ed. "Sequence A036691 (Compositorial numbers: product of first n composite numbers.)". OEIS Foundation. https://oeis.org/A036691. Retrieved 2023-04-03. 
  5. ↑ Sloane, N. J. A., ed. "Sequence A001597 (Perfect powers)". OEIS Foundation. https://oeis.org/A001597. Retrieved 2023-04-03. 
  6. ↑ Sloane, N. J. A., ed. "Sequence A005934 (Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization)". OEIS Foundation. https://oeis.org/A005934. Retrieved 2023-04-13. 
  7. ↑ Sloane, N. J. A., ed. "Sequence A005361 (Product of exponents of prime factorization of n.)". OEIS Foundation. https://oeis.org/A005361. Retrieved 2023-04-13. 
  8. ↑ Sloane, N. J. A., ed. "Sequence A001013 (Jordan-Polya numbers: products of factorial numbers)". OEIS Foundation. https://oeis.org/A001013. Retrieved 2023-04-03. 
  9. ↑ "1728". https://www.numbersaplenty.com/1728. 
  10. ↑ Sloane, N. J. A., ed. "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and relatively prime to n)". OEIS Foundation. https://oeis.org/A000010. Retrieved 2023-04-03. 
  11. ↑ Sloane, N. J. A., ed. "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". OEIS Foundation. https://oeis.org/A005101. Retrieved 2023-04-03. 
  12. ↑ Sloane, N. J. A., ed. "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". OEIS Foundation. https://oeis.org/A005835. Retrieved 2023-04-03. 
  13. ↑ Sloane, N. J. A., ed. "Sequence A005153 (Practical numbers)". OEIS Foundation. https://oeis.org/A005153. Retrieved 2023-04-03. 
  14. ↑ Sloane, N. J. A., ed. "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". OEIS Foundation. https://oeis.org/A083207. Retrieved 2023-04-03. 
  15. ↑ Sloane, N. J. A., ed. "Sequence A003586 (3-smooth numbers: numbers of the form 2^i*3^j with i, j greater than or equal to 0.)". OEIS Foundation. https://oeis.org/A003586. Retrieved 2023-04-04. 
  16. ↑ Sloane, N. J. A., ed. "Sequence A051037 (5-smooth numbers)". OEIS Foundation. https://oeis.org/A051037. Retrieved 2023-04-04. 
    Equivalently, regular numbers.
  17. ↑ Sloane, N. J. A., ed. "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". OEIS Foundation. https://oeis.org/A000005. Retrieved 2023-04-04. 
  18. ↑ Sloane, N. J. A., ed. "Sequence A005114 (Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function.)". OEIS Foundation. https://oeis.org/A005114. Retrieved 2023-04-03. 
  19. ↑ 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. 
  20. ↑ John McKay (2001). "The Essentials of Monstrous Moonshine". Groups and Combinatorics: In memory of Michio Suzuki. Advanced Studies in Pure Mathematics. 32. Tokyo: Mathematical Society of Japan. p. 351. doi:10.2969/aspm/03210347. ISBN 978-4-931469-82-2. 
  21. ↑ Sloane, N. J. A., ed. "Sequence A165134 (Number of directed Hamiltonian paths in the n X n knight graph)". OEIS Foundation. https://oeis.org/A165134. Retrieved 2022-11-30. 
  22. ↑ 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)". OEIS Foundation. https://oeis.org/A011541. Retrieved 2022-11-30. 
  23. ↑ Śrī Dharmavira Prabhu. "Chanting 64 rounds Harināma daily!" (in en-US). Śrī Gaura Radha Govinda International. https://www.dharmavira.com/64-rounds-harinam/. 

External links



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