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The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base. [ 60 ] Another result on divisibility of factorials, Wilson's theorem , states that ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1} is divisible by n {\displaystyle n} if and only if n {\displaystyle n} is a prime number . [ 52 ]
This approximation is good to more than 8 decimal digits for z with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.
Let be a natural number. For a base >, we define the sum of the factorials of the digits [5] [6] of , :, to be the following: = =!. where = ⌊ ⌋ + is the number of digits in the number in base , ! is the factorial of and
Numbers like zero don't fit this criterion. They should be easily recognized by the programmer as invalid values in the debugger. On machines that don't have byte alignment, they should be odd numbers, so that dereferencing them as addresses causes an exception. They should cause an exception, or perhaps even a debugger break, if executed as code.
The final expression is defined for all complex numbers except the negative even integers and satisfies (z + 2)!! = (z + 2) · z!! everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in the sense of the Bohr–Mollerup theorem.
But if exact values for large factorials are desired, then special software is required, as in the pseudocode that follows, which implements the classic algorithm to calculate 1, 1×2, 1×2×3, 1×2×3×4, etc. the successive factorial numbers. constants: Limit = 1000 % Sufficient digits.
Here the fibonorial constant (also called the fibonacci factorial constant [1]) is defined by = = (), where = and is the golden ratio. An approximate truncated value of C {\displaystyle C} is 1.226742010720 (see (sequence A062073 in the OEIS ) for more digits).