I was watching flammable maths' video on ambiguous math trolls, specifically one involving ratios (No, not the twitter ratios) and factorials. He stated:
25 - 5 : 5 = 4!
721 - 103 : 103 = 6!
Where the ! mark at the end could either be the factorial symbol or just an exclamation mark. He then derived:
For a - b : b = k! (k! in the sense of the riddle)
a = k! + 1 (real factorial this time)
b = (k! + 1)/(k + 1) [again, real factorial]
He then concluded that k must be even since if it were odd then we would wind up with odd/even which isn't a whole number.
By similar logic I found out that k+1 must be prime because if it weren't then k!+1 wouldn't divide k+1.
I did some experimenting and tested prime k+1 until 19 and it turns out all the prime values of k+1 work.
So my question is: do all prime numbers P follow the property that p divides (p-1)! + 1?