288 (number)

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title: "288 (number)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integers"] topic_path: "general/integers" source: "https://en.wikipedia.org/wiki/288_(number)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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::callout[type=note] the number 288 ::

| number = 288 288 (two hundred [and] eighty-eight) is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".

In mathematics

Factorization properties

Because its prime factorization 288 = 2^5\cdot 3^2 contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number. This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization.{{cite journal | last1 = Hardy | first1 = G. E. | last2 = Subbarao | first2 = M. V. | author2-link = Mathukumalli V. Subbarao | journal = Congressus Numerantium | mr = 703589 | pages = 277–307 | title = Highly powerful numbers | url = https://oeis.org/A005934/a005934.pdf | volume = 37 | year = 1983}} Among the highly abundant numbers, numbers with record-setting sums of divisors, it is one of only 13 such numbers with an odd divisor sum.

Both 288 and are powerful numbers, numbers in which all exponents of the prime factorization are larger than one. This property is closely connected to being highly abundant with an odd divisor sum: all sufficiently large highly abundant numbers have an odd prime factor with exponent one, causing their divisor sum to be even.{{cite journal | last1 = Alaoglu | first1 = L. | author1-link = Leonidas Alaoglu | last2 = Erdős | first2 = P. | author2-link = Paul Erdős | doi = 10.2307/1990319 | journal = Transactions of the American Mathematical Society | jstor = 1990319 | mr = 11087 | pages = 448–469 | title = On highly composite and similar numbers | url = https://www.renyi.hu/~p_erdos/1944-03.pdf | volume = 56 | year = 1944| issue = 3 }} 288 and 289 form only the second consecutive pair of powerful numbers after {{nowrap|8 and 9.{{cite book | last = De Koninck | first = Jean-Marie | author-link = Jean-Marie De Koninck | doi = 10.1090/mbk/064 | isbn = 978-0-8218-4807-4 | mr = 2532459 | page = 69 | publisher = American Mathematical Society | location = Providence, Rhode Island | title = Those fascinating numbers | url = https://books.google.com/books?id=xdfTAwAAQBAJ&pg=PA69 | year = 2009}}}}

Factorial properties

288 is a superfactorial, a product of consecutive factorials, since{{cite book | last = Wells | first = David | isbn = 9780140261493 | page = 137 | publisher = Penguin | title = The Penguin Dictionary of Curious and Interesting Numbers | url = https://books.google.com/books?id=kQRPkTkk_VIC&pg=PA137 | year = 1997}}{{cite journal | last1 = Kozen | first1 = Dexter | author1-link = Dexter Kozen | last2 = Silva | first2 = Alexandra | author2-link = Alexandra Silva | doi = 10.4169/amer.math.monthly.120.02.131 | issue = 2 | journal = The American Mathematical Monthly | jstor = 10.4169/amer.math.monthly.120.02.131 | mr = 3029938 | pages = 131–139 | title = On Moessner's theorem | volume = 120 | year = 2013| hdl = 2066/111198 | s2cid = 8799795 | hdl-access = free

288 appears prominently in Stirling's approximation for the factorial, as the denominator of the second term of the Stirling series n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +\frac{1}{12n}+\frac{1}{288n^2} - \frac{139}{51840n^3} -\frac{571}{2488320n^4}+ \cdots \right).

Figurate properties

288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number and a dodecagonal number.{{cite book | last1 = Deza | first1 = Elena | author1-link = Elena Deza | last2 = Deza | first2 = Michel | author2-link = Michel Deza | isbn = 9789814355483 | pages = 3, 23, 211 | publisher = World Scientific | title = Figurate Numbers | year = 2012}} Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number: 41616 = \frac{288\cdot 289}{2} = 204^2.

Enumerative properties

There are 288 different ways of completely filling in a 4\times 4 sudoku puzzle grid.{{cite journal | last = Taalman | first = Laura | author-link = Laura Taalman | date = September 2007 | doi = 10.1080/10724117.2007.11974720 | issue = 1 | journal = Math Horizons | jstor = 25678701 | pages = 5–9 | title = Taking Sudoku seriously | volume = 15| s2cid = 126371771 }} For square grids whose side length is the square of a prime number, such as 4 or 9, a completed sudoku puzzle is the same thing as a "pluperfect Latin square", an n\times n array in which every dissection into n rectangles of equal width and height to each other has one copy of each digit in each rectangle. Therefore, there are also 288 pluperfect Latin squares of order 4. There are 288 different 2\times 2 invertible matrices modulo six, and 288 different ways of placing two chess queens on a 6\times 6 board with toroidal boundary conditions so that they do not attack each other. There are 288 independent sets in a 5-dimensional hypercube, up to symmetries of the hypercube.

In other areas

In early 20th-century molecular biology, some mysticism surrounded the use of 288 to count protein structures, largely based on the fact that it is a smooth number.{{cite journal | last = Potter | first = Robert D. | date = February 12, 1938 | doi = 10.2307/3914385 | issue = 7 | journal = The Science News-Letter | jstor = 3914385 | pages = 99–100 | title = Building blocks of life ruled by the number 288: This number and its multiples found everywhere in groupings of amino acids to form proteins | volume = 33}}{{cite journal | last = Klotz | first = Irving M. | date = October 1993 | doi = 10.1096/fasebj.7.13.8405807 | issue = 13 | journal = The FASEB Journal | pages = 1219–1225 | title = Biogenesis: number mysticism in protein thinking | volume = 7| pmid = 8405807 | s2cid = 13276657 | doi-access = free

A common mathematical pun involves the fact that and that 144 is named as a gross: "Q: Why should the number 288 never be mentioned? A: it is two gross."{{cite book | last = Nowlan | first = Robert A. | contribution = Logical Nonsense | doi = 10.1007/978-94-6300-893-8_17 | pages = 263–268 | publisher = Sense Publishers | title = Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them | year = 2017| isbn = 978-94-6300-893-8

References

References

  1. {{cite OEIS. A003586. 3-smooth numbers
  2. {{cite OEIS. A005934. Highly powerful numbers
  3. {{cite OEIS. A128700. Highly abundant numbers with an odd divisor sum
  4. A060355. Numbers n such that n and n+1 are a pair of consecutive powerful numbers
  5. {{cite OEIS. A000178. Superfactorials
  6. {{cite OEIS. A001923
  7. {{cite OEIS. A001164. Stirling's formula: denominators of asymptotic series for Gamma function
  8. {{cite OEIS. A002411. Pentagonal pyramidal numbers
  9. {{cite OEIS. A051624. 12-gonal (or dodecagonal) numbers
  10. {{cite OEIS. A001108. a(n)-th triangular number is a square
  11. {{cite OEIS. A107739. Number of (completed) sudokus (or Sudokus) of size n^2 X n^2
  12. {{cite OEIS. A108395. Number of pluperfect Latin squares of order n
  13. {{cite OEIS. A000252. Number of invertible 2 X 2 matrices mod n
  14. {{cite OEIS. A172517. Number of ways to place 2 nonattacking queens on an n X n toroidal board
  15. {{cite OEIS. A060631. Number of independent sets in an n-dimensional hypercube modulo symmetries of the hypercube

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integers