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A power of two is a number of the form 2 n where ... but they are often the sum or product of only two or three powers of two, or powers of two minus one. For example
Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2 n members. Integer powers of 2 are important in computer science. The positive integer powers 2 n give the number of possible values for an n-bit integer binary number; for example, a byte may take 2 8 = 256 different values.
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. [1] [2] The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition): x % 2 n == x & (2 n - 1) Examples: x % 2 == x & 1 x % 4 == x & 3 x % 8 == x & 7
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Since a = n(n + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in n having factors n 2 and (n + 1) 2, while for an even power the polynomial has factors n, n + 1/2 and n + 1.