second time puzzled.

(p-1)! + 1 is not a power for p for prime p > 5.

Couple of weekends back , I was bored and was aching to get out of the drudge of bug fixing. I was going through the exercises in Niven's chapter on congruences looking for some problem that would make my day. I picked this one as it seemed hard but was not marked hard in the exercises. It took two weekends to bring this problem down. At this rate i am worried I would never would finish niven's book.

But first a proof for the case p = 7, which is simple

let (p-1)! +1 = p^a

We know that 6! is divisible by 16. The first step is the hardest but how it hides the vain attempts before this.

We must have
7^a = 1 mod 16

We also know that since 7^2 = 1 mod 16 and 2 is the least
such positive power, `a' must be a multiple of 2.

6! is divisible by 9

7^a = 1 mod 9
but 7^3 = 1 mod 9 ( 3 being the least such power)

a is a multiple of 2 and of 3
a is a multiple of the lcm (2 , 3) = 6

but makes 7^a too large to be equal to 6! + 1.

The next post will contain the proof for p > 7 .( not because of lack of margin space in this post :-)