(In reply to

re: spoiler by Jer)

Great proof, xdog. I disagree with Jer. Your proof is complete and correct even if there does not exist an S that meets all conditions.

Similarly, I have discovered a truly marvelous proof that if x = 1 and x = 2, then 1 = 2. (Unfortunately, the proof is too short to fit in this space.) The proof is complete and valid, even though no x exists that meets the initial conditions. That turns out to be fortunate, as I would not want to live in a world where 1 = 2.

But back to S. Here is another set that works:

(a,b,c,d,e,f) = (6,6,6,6,6,6) and S = 36. As xdog proved, 36 must be composite.

*Edited on ***February 18, 2013, 10:23 am**