Monday, 14 December 2015

Alternating sums of factorials

The first six alternating sums of factorials

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

are all primes.  The seventh

9! − 8! + 7! − 6! + 5! − 4! + 3! − 2! + 1! = 326981 = 79 × 4139

is not.

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