The Bohr Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis.
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As an example, the function argument in the recursive expression for the factorial function below will always decrease by 1; from the well-ordering property on natural numbers, the argument will eventually reach 1 and the recursion will terminate.
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In mathematics, and more particularly in number theory, "'primorial "'is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, only prime numbers are multiplied.
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To assure that the exact result will fit for all legal values of even the smallest commonly used integral type ( 8-bit signed integers ) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow.
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The same function can be expressed with clausal function definitions where the " if "-" then "-" else " conditional is replaced by a sequence of templates of the factorial function evaluated for specific values, separated by'|', which are tried one by one in the order written until a match is found:
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A single-argument version A ( k ) = A ( k, k ) that increases both " m " and " n " at the same time dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).
