Determine all possible octuplets (A, B, C, D, E, F, G, H) of positive integers, with A ≤ B ≤ C ≤ D, and, E ≤ F ≤ G ≤ H and, A ≤ E, that satisfy both the equations: A+B+C+D = E*F*G*H and, A*B*C*D = E+F+G+H.

Prove that these are the only octuplets that exist.

(In reply to

Mutually inclusive by brianjn)

I think I was considering all of the criteria when I posed my two additional cases: in each, the two sets of four increase monotonically; A is equal to E (so is "equal or less than E"); and the additions and products work in both directions for both cases. In what sense do my examples have "equations that stand alone" ? If ABCD is the "left side" and EFGH the "right side" both cases meet both tests (sum of ABCD = product of EFGH) and (product of ABCD = sum of EFGH).

I was, however, overly optimistic in claiming "and others" since I have not yet found any more than these three cases. It almost seems that the A,B,E,and F must all be "1", but I have no argument for that, other than the empirical testing. The original challenge was to "prove that these are the only octuplets that exist" and I certainly have not done that. My programs did turn up Ady's case as well as the other two.

I'll be away for the long weekend, but will check in next Monday to see where the discussion stands, and especially to see if any proof (other than exhaustive enumeration) has been given.