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This math.se comment by Daniel Wainfleet offers a problem that I found fun.
M. Wainfleet quotes the author (Rouse Ball or Coxeter):
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!!.
