In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an alternating factorial is the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the
alternating sum of the first ''n''
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
s of
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
s.
This is the same as their sum, with the
odd-indexed factorials multiplied by
−1
In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.
...
if ''n'' is
even, and the even-indexed factorials multiplied by −1 if ''n'' is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically,
:
or with the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
in which af(1) = 1.
The first few alternating factorials are
:
1, 1,
5,
19,
101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019
For example, the third alternating factorial is 1! – 2! + 3!. The fourth alternating factorial is −1! + 2! − 3! + 4! = 19. Regardless of the
parity of ''n'', the last (''n''th) summand, ''n''!, is given a positive sign, the (''n'' – 1)th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly.
This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of ''n'') changes the signs of the resulting sums but not their absolute values.
proved that there are only a finite number of alternating factorials that are also
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, since 3612703
divides
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
af(3612702) and therefore divides af(''n'') for all ''n'' ≥ 3612702. The primes are af(''n'') for
:''n'' = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, ...
with several higher
probable prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific co ...
s that have not been proven prime.
Notes
References
*
*
* Yves Gallot
\sum_^{n - 1} i! finite?">Is the number of primes finite?* Paul Jobling
Guy's problem B43: search for primes of form n!-(n-1)!+(n-2)!-(n-3)!+...+/-1!
Integer sequences
Factorial and binomial topics