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If m is a power of 2, then a − 1 should be divisible by 4 but not divisible by 8, i.e. a ≡ 5 (mod 8). [1]: §3.2.1.3 Indeed, most multipliers produce a sequence which fails one test for non-randomness or another, and finding a multiplier which is satisfactory to all applicable criteria [1]: §3.3.3 is quite challenging. [8]
x 1 = x; x 2 = x 2 for i = k - 2 to 0 do if n i = 0 then x 2 = x 1 * x 2; x 1 = x 1 2 else x 1 = x 1 * x 2; x 2 = x 2 2 return x 1. The algorithm performs a fixed sequence of operations (up to log n): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value. A similar algorithm for ...
In mathematics and computer science, Recamán's sequence [1] [2] is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.
is constant-recursive because it satisfies the linear recurrence = +: each number in the sequence is the sum of the previous two. [2] Other examples include the power of two sequence ,,,,, …, where each number is the sum of twice the previous number, and the square number sequence ,,,,, ….
[1] [2] Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. [3] The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite ...
Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2 k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k, which is n c where c = log 2 3.
The problem here is that the low-order bits of a linear congruential PRNG with modulo 2 e are less random than the high-order ones: [6] the low n bits of the generator themselves have a period of at most 2 n. When the divisor is a power of two, taking the remainder essentially means throwing away the high-order bits, such that one ends up with ...
Other examples of total recursive but not primitive recursive functions are known: The function that takes m to Ackermann(m,m) is a unary total recursive function that is not primitive recursive. The Paris–Harrington theorem involves a total recursive function that is not primitive recursive. The Sudan function; The Goodstein function