Content-Type: text/shitpost

This math.se comment by Daniel Wainfleet offers a problem that I found fun.

Let !!p,q!! be consecutive primes greater than 2. Show that !!p+q!! is a product of three integers, each greater than 1.

M. Wainfleet quotes the author (Rouse Ball or Coxeter):

At first glance this seems very hard. In fact it is quite easy.

This is a very useful hint!

Mouse over to view the solution:

Since !!p!! and !!q!! are both odd, their mean, !!\frac12(p+q)!!, is an integer. It cannot be prime, since !!p!! and !!q!! are consecutive primes, so !!\frac12(p+q) = ab!! for some !!a!! and !!b!! each greater than 1. Then !!p+q = 2ab!!.