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288 (number)

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".

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.^[1] 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.^[2][3] 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.^[4]

Both 288 and 289 = 17² are powerful numbers, numbers in which all exponents of the prime factorization are larger than one.^[5][6][7] 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.^[4][8] 288 and 289 form only the second consecutive pair of powerful numbers after 8 and 9.^[5][6][7]

288 is a superfactorial, a product of consecutive factorials, since^[5][9][10] $${\displaystyle 288=1!\cdot 2!\cdot 3!\cdot 4!=1^{4}\cdot 2^{3}\cdot 3^{2}\cdot 4^{1}.}$$ Coincidentally, as well as being a product of descending powers, 288 is a sum of ascending powers:^[11] $${\displaystyle 288=1^{1}+2^{2}+3^{3}+4^{4}.}$$

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

288 is connected to the figurate numbers in multiple ways. It is a pentagonal pyramidal number^[13][14] and a dodecagonal number.^[14][15] Additionally, it is the index, in the sequence of triangular numbers, of the fifth square triangular number:^[14][16] $${\displaystyle 41616={\frac {288\cdot 289}{2}}=204^{2}.}$$

There are 288 different ways of completely filling in a ${\displaystyle 4\times 4}$ sudoku puzzle grid.^[17][18] 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.^[19] There are 288 different ${\displaystyle 2\times 2}$ invertible matrices modulo six,^[20] 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.^[21] There are 288 independent sets in a 5-dimensional hypercube, up to symmetries of the hypercube.^[22]

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.^[23][24]

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."^[25]