Show that if k is any positive integer, then the numbers (k+1)! + j, where j = 2,3,...,k+1, are all composite.
I believe this is essentially saying that there exists arbitrarily long sequences of consecutive composite numbers. I am just having a little trouble with the proof. Any suggestions would be great.
Given that (k + 1)! = 1(2)(3)(4)...(k - 1)k(k + 1), it's evident that every positive integer is covered in that factorial up through k + 1. So no matter what j is, you can evenly divide j into (k + 1)! (and obviously j evenly divides j) leaving (k + 1)!/j + 1 from the given problem.