Set partitions

[Risto Kauppila]

For finite sets A of integers, let P(A) be the product of integers
in A.
If { X_1, Y_1 } is a partition of a finite set S of positive distinct
integers such that
p(X_1) + p(Y_1) is the minimum of p(X) + p(Y) over all partitions of S
into non-empty sets X and Y, does that partition also minimize
|p(X) - p(Y)|?
I'm especially interested in the case, where S is the first
n primes.

[quasi]

On Wed, 09 Feb 2011 21:54:18 +0200, Risto Kauppila
<risto.k...@NOSPAM.saunalahti.fi.invalid> wrote:

>For finite sets A of integers, let P(A) be the product of
>integers in A.
>
>If { X_1, Y_1 } is a partition of a finite set S of positive
>distinct integers such that p(X_1) + p(Y_1) is the minimum of
>p(X) + p(Y) over all partitions of S into non-empty sets X and Y,
>does that partition also minimize |p(X) - p(Y)|?

Yes, since P(X)*P(Y) is constant, minimizing p(X) + p(Y) also
minimizes |p(X) - p(Y)|.

This follows easily from the identity

(p(X) - p(y))^2 = (p(X) + p(Y)^2 - 4*p(X)*p(Y).

quasi

[Risto Kauppila]

Or (p(X) + p(Y))^2 = p(X)^2 + p(Y)^2 + 2p(X)p(Y) is minimum,
when p(X)^2 + p(Y)^2 is minimum, since 2p(X)p(Y) is constant.
And then also (p(X) - p(Y))^2 = p(X)^2 + p(Y)^2 - 2p(X)p(Y)
is minimum.

Thanks