288 (two hundred
ndeighty-eight) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
following
287
Year 287 (Roman numerals, CCLXXXVII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Diocletian and Maximian (or, less frequ ...
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 contains only the first two
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s 2 and 3, 288 is a
3-smooth number. This factorization also makes it a
highly powerful number In elementary number theory, a highly powerful number is a positive integer that satisfies a property introduced by the Indo-Canadian mathematician Mathukumalli V. Subbarao. The set of highly powerful numbers is a proper subset of the set of powerf ...
, a number with a record-setting value of the product of the exponents in its factorization. Among the
highly abundant number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar cla ...
s, 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 number
A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a ...
s, 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
]
Factorial properties
288 is a superfactorial, a product of consecutive factorials, since Coincidentally, as well as being a product of descending powers, 288 is a sum of ascending powers:
288 appears prominently in Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
for the factorial, as the denominator of the second term of the Stirling series
Figurate properties
288 is connected to the figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polyg ...
s in multiple ways. It is a pentagonal pyramidal number A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. ...
[ and a ]dodecagonal number
A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for ''n'' is given by the formula
:D_=5n^2 - 4n
The first few dodecagonal numbers are:
: 0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 7 ...
. Additionally, it is the index, in the sequence of triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s, of the fifth square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are:
:0, 1, 36, , , , , , ,
Expl ...
:[
]
Enumerative properties
There are 288 different ways of completely filling in a sudoku
Sudoku (; ja, 数独, sūdoku, digit-single; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row ...
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 array in which every dissection into 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 invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
modulo six, and 288 different ways of placing two chess queens on a 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
Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and physi ...
, some mysticism surrounded the use of 288 to count protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
structures, largely based on the fact that it is a smooth number.
A common mathematical pun
A pun, also known as paronomasia, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect. These ambiguities can arise from the intentional use of homophoni ...
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."[ See p. 284.]
References
{{Integers, 2
Integers