Show that the sum of two consecutive odd primes has at least 3 (not necessarily distinct) prime factors. For example,

3+5=2*2*2
5+7=2*2*3
7+11=2*3*3

If P and Q are consecutive odd primes, then their average ½(P+Q) is integer, and not a prime since P<½(P+Q)<Q, so ½(P+Q)=MN, and then P+Q=2MN, as desired.

Posted by Old Original Oskar! on 2005-01-24 13:56:19