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  2. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    This logarithmic number of operations is to be compared with the trivial algorithm which requires n − 1 multiplications. This algorithm is not tail-recursive. This implies that it requires an amount of auxiliary memory that is roughly proportional to the number of recursive calls -- or perhaps higher if the amount of data per iteration is ...

  3. Recursion (computer science) - Wikipedia

    en.wikipedia.org/wiki/Recursion_(computer_science)

    [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 ...

  4. Recursion - Wikipedia

    en.wikipedia.org/wiki/Recursion

    For example, the formal definition of the natural numbers by the Peano axioms can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number." [ 2 ] By this base case and recursive rule, one can generate the set of all natural numbers.

  5. Primitive recursive function - Wikipedia

    en.wikipedia.org/wiki/Primitive_recursive_function

    A total recursive function is a partial recursive function that is defined for every input. Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known example of a total recursive function (in fact, provable total), that is not primitive ...

  6. Exponentiation - Wikipedia

    en.wikipedia.org/wiki/Exponentiation

    The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and ...

  7. General recursive function - Wikipedia

    en.wikipedia.org/wiki/General_recursive_function

    The number e is called an index or Gödel number for the function f. [10]: 52–53 A consequence of this result is that any μ-recursive function can be defined using a single instance of the μ operator applied to a (total) primitive recursive function.

  8. Modular exponentiation - Wikipedia

    en.wikipedia.org/wiki/Modular_exponentiation

    Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m .

  9. Ackermann function - Wikipedia

    en.wikipedia.org/wiki/Ackermann_function

    First, place the natural numbers along the top row. To determine a number in the table, take the number immediately to the left. Then use that number to look up the required number in the column given by that number and one row up. If there is no number to its left, simply look at the column headed "1" in the previous row.