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A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 (2 5). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of
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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.
The clique problem and the independent set problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa. [20] Therefore, many computational results may be applied equally well to either problem, and some research papers do not clearly distinguish between the two problems.
If n is a small fixed number, then an exhaustive search for the solution is practical. L - the precision of the problem, stated as the number of binary place values that it takes to state the problem. If L is a small fixed number, then there are dynamic programming algorithms that can solve it exactly. As both n and L grow large, SSP is NP-hard.
Thus, non-trapezoidal polite number must have the form of a power of two multiplied by an odd prime. As Jones and Lord observe, [12] there are exactly two types of triangular numbers with this form: the even perfect numbers 2 n − 1 (2 n − 1) formed by the product of a Mersenne prime 2 n − 1 with half the nearest power of two, and
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For i = 0, 1, 2, ..., the algorithm compares x 2 i −1 with each subsequent sequence value up to the next power of two, stopping when it finds a match. It has two advantages compared to the tortoise and hare algorithm: it finds the correct length λ of the cycle directly, rather than needing to search for it in a subsequent stage, and its ...