Alternating sums of factorials

Richard Guy’s Strong Law of Small Numbers says

There aren’t enough small numbers to meet the many demands made of them.

In his article by the same name [1] Guy illustrates his law with several examples of patterns that hold for small numbers but eventually fail. One of these examples is

3! – 2! + 1! = 5

4! – 3! + 2! – 1! = 19

5! – 4! + 3! – 2! + 1! = 101

6! – 5! + 4! – 3! + 2! – 1! = 619

7! – 6! + 5! – 4! + 3! – 2! + 1! = 4421

8! – 7! + 6! – 5! + 4! – 3! + 2! – 1! = 35899

If we let f(n) be the alternating factorial sum starting with n, f(n) is prime for n = 3, 4, 5, 6, 7, 8, but not for n = 9. So the alternating sums aren’t all prime. Is f(n) usually prime? f(10) is, so maybe 9 is the odd one. Let’s write a code to find out.

    from sympy import factorial, isprime

    def altfact(n):
        sign = 1
        sum = 0
        while n > 0:
            sum += sign*factorial(n)
            sign *= -1
            n -= 1
        return sum

    numprimes = 0
    for i in range(3, 1000):
        if isprime( altfact(i) ):
            print(i)
            numprimes += 1
    print(numprimes)

You could speed up this code by noticing that

    altfact(n+1) = factorial(n+1) - altfact(n)

and tabulating the values of altfact. The code above corresponds directly to the math, though it takes a little while to run.

So it turns out the alternating factorial sum is only prime for 15 values less than 1000. In addition to the values of n mentioned above, the other values are 15, 19, 41, 59, 61, 105, 160, and 601.

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[1] The Strong Law of Small Numbers, Richard K. Guy, The American Mathematical Monthly, Vol. 95, No. 8 (Oct., 1988), pp. 697-712.