# Composite Numbers

• Feb 2nd 2010, 04:34 AM
Composite Numbers
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.

Thanks!
• Feb 2nd 2010, 06:47 AM
aman_cc
Quote:

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.

Thanks!

Simply noting that
j|[(k+1)! + j] , for all j = 2,3,...,k+1
should do I guess.

Or I am not getting it?
• Feb 2nd 2010, 03:42 PM
wonderboy1953
Would this be a good argument?
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.