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The special case of Legendre's formula for = gives the number of trailing zeros in the decimal representation of the factorials. [57] According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of from , and dividing the result by four. [58]
Now the function + is unimodal, with maximum value zero. Locally around zero, it looks like − t 2 / 2 {\displaystyle -t^{2}/2} , which is why we are able to perform Laplace's method. In order to extend Laplace's method to higher orders, we perform another change of variables by 1 + t − e t = − τ 2 / 2 {\displaystyle 1+t-e^{t}=-\tau ^{2}/2} .
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
A googol is the large number 10 100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 10, 000, 000 ...
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
The ring of 2×2 matrices with integer entries does not satisfy the zero-product property: if = and = (), then = () = =, yet neither nor is zero. The ring of all functions: [,], from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the ...
More recent results providing Jacobi-type J-fractions that generate the single factorial function and generalized factorial-related products lead to other new congruence results for the Stirling numbers of the first kind. [13] For example, working modulo we can prove that
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.