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Factors and Primes (Posted on 2007-01-24)

Prove that n!-1 is a composite number when n>3 and n+2 is a prime.

No Solution Yet Submitted by atheron

Rating: 4.6667 (3 votes)

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Puzzle Solution

Comment 4 of 4 |

Let m=n+2

If m is a prime number, then in terms of Wilson's Theorem we must have:

(m-1)! == - 1 (mod m)

=> (m-2)! == 1 (mod m)

=> (m-2)! -1 == 0 (mod m)

So, if m is a prime number, then (m-2)! - 1 is divisible by m.

Recalling that m= n+2, we now observe that:

(m-2)!-1 or, n!-1 is a composite number.

Posted by K Sengupta on 2022-07-11 23:47:15

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