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288 (two hundred 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 $${\displaystyle 288=2^{5}\cdot 3^{2}}$$ 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. 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 289 = 17 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. 288 and 289 form only the second consecutive pair of powerful numbers after 8 and 9.

Factorial properties

288 is a superfactorial, a product of consecutive factorials, since $${\displaystyle 288=1!\cdot 2!\cdot 3!\cdot 4!=1^{4}\cdot 2^{3}\cdot 3^{2}\cdot 4^{1}.}$$ Coincidentally, as well as being a product of descending powers, 288 is a sum of ascending powers: $${\displaystyle 288=1^{1}+2^{2}+3^{3}+4^{4}.}$$

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

Figurate properties

288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number and a dodecagonal number. Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number: $${\displaystyle 41616={\frac {288\cdot 289}{2}}=204^{2}.}$$

Enumerative properties

There are 288 different ways of completely filling in a ${\displaystyle 4\times 4}$ sudoku puzzle grid. 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 ${\displaystyle n\times n}$ array in which every dissection into ${\displaystyle n}$ 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 ${\displaystyle 2\times 2}$ invertible matrices modulo six, and 288 different ways of placing two chess queens on a ${\displaystyle 6\times 6}$ 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.

A common mathematical pun involves the fact that 288 = 2 · 144, and that 144 is named as a gross: "Q: Why should the number 288 never be mentioned? A: it is two gross."

References

1. Sloane, N. J. A. (ed.). "Sequence A003586 (3-smooth numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Sloane, N. J. A. (ed.). "Sequence A005934 (Highly powerful numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Hardy, G. E.; Subbarao, M. V. (1983). "Highly powerful numbers" (PDF). Congressus Numerantium. 37: 277–307. MR 0703589.
4. ^ Sloane, N. J. A. (ed.). "Sequence A128700 (Highly abundant numbers with an odd divisor sum)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. ^ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers. Penguin. p. 137. ISBN 9780140261493.
6. ^ Sloane, N. J. A. (ed.). "Sequence A060355 (Numbers n such that n and n+1 are a pair of consecutive powerful numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. ^ De Koninck, Jean-Marie (2009). Those fascinating numbers. Providence, Rhode Island: American Mathematical Society. p. 69. doi:10.1090/mbk/064. ISBN 978-0-8218-4807-4. MR 2532459.
8. Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
9. Sloane, N. J. A. (ed.). "Sequence A000178 (Superfactorials)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. Kozen, Dexter; Silva, Alexandra (2013). "On Moessner's theorem". The American Mathematical Monthly. 120 (2): 131–139. doi:10.4169/amer.math.monthly.120.02.131. hdl:2066/111198. JSTOR 10.4169/amer.math.monthly.120.02.131. MR 3029938. S2CID 8799795.
11. Sloane, N. J. A. (ed.). "Sequence A001923". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
12. Sloane, N. J. A. (ed.). "Sequence A001164 (Stirling's formula: denominators of asymptotic series for Gamma function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
13. Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. ^ Deza, Elena; Deza, Michel (2012). Figurate Numbers. World Scientific. pp. 3, 23, 211. ISBN 9789814355483.
15. Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
16. Sloane, N. J. A. (ed.). "Sequence A001108 (a(n)-th triangular number is a square)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
17. Sloane, N. J. A. (ed.). "Sequence A107739 (Number of (completed) sudokus (or Sudokus) of size n^2 X n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
18. Taalman, Laura (September 2007). "Taking Sudoku seriously". Math Horizons. 15 (1): 5–9. doi:10.1080/10724117.2007.11974720. JSTOR 25678701. S2CID 126371771.
19. Sloane, N. J. A. (ed.). "Sequence A108395 (Number of pluperfect Latin squares of order n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
20. Sloane, N. J. A. (ed.). "Sequence A000252 (Number of invertible 2 X 2 matrices mod n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
21. Sloane, N. J. A. (ed.). "Sequence A172517 (Number of ways to place 2 nonattacking queens on an n X n toroidal board)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
22. Sloane, N. J. A. (ed.). "Sequence A060631 (Number of independent sets in an n-dimensional hypercube modulo symmetries of the hypercube)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
23. Potter, Robert D. (February 12, 1938). "Building blocks of life ruled by the number 288: This number and its multiples found everywhere in groupings of amino acids to form proteins". The Science News-Letter. 33 (7): 99–100. doi:10.2307/3914385. JSTOR 3914385.
24. Klotz, Irving M. (October 1993). "Biogenesis: number mysticism in protein thinking". The FASEB Journal. 7 (13): 1219–1225. doi:10.1096/fasebj.7.13.8405807. PMID 8405807. S2CID 13276657.
25. Nowlan, Robert A. (2017). "Logical Nonsense". Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them. Sense Publishers. pp. 263–268. doi:10.1007/978-94-6300-893-8_17. ISBN 978-94-6300-893-8. See p. 284.

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