While messing around with factorials, I noticed this:
$$3! - 2! + 1! = 6 - 2 + 1 = 5$$ $$4! - 3! + 2! - 1!= 24 - 6 + 2 - 1=19$$ $$5! - 4! + 3! - 2! + 1! = 5! - 19 = 101$$ $$6! - 5! + 4! - 3! + 2! - 1! = 6! - 101 = 619$$ $$7! - 6! + 5! - 4! + 3! - 2! + 1! = 7! - 619 = 4421$$
Notice that all of the sums are prime.
My question, is, does this pattern (i.e. the results are prime) continue forever? (If yes, please prove why. If no, please provide a counterexample, and why it happens.)
My attempt was to show that for every positive integer $n > 2$, $(n - 1)! - (n + 2)! + (n + 3)! - (n + 4)! $ (and so on) $ = k$, and prove that $k $ is not divisible by any prime below or equal to $(n - 1)$. (Which by the way, I almost proved it. Will edit this post if I did it.)
But, I am really confused on how to show that $k$ is not divisible by any prime bigger than $(n - 1).$ Is my approach to solve this problem correct?
No, this pattern does not continue forever. Quite the opposite - there exist finitely many primes of this form. See https://oeis.org/A071828 and the reference there.