Unique sum sets

thabit

A finite set $S=\{a_1,...,a_n\}$ positive integers is called a unique sum set if for each sequence x1,...,xn of nonnegative integers the following implication holds:
$a_1x_1+\cdots+a_nx_n=a_1+\cdots+a_n \to x_1=\ldots=x_n=1.$
In other words, if the only way to write the sum of all elements of $S$ as a sum of elements of $S$ , where we may use some elements more than once, is the sum of all elements itself.

First an easy question:
show that
$\{2^n-2^0, 2^n-2^1,\ldots,2^n-2^{n-1}\}$
is a unique sum set for each $n$ .

Now a challenging one:
is the unique sum set given in the previous question the one with the smallest sum if the number of elements is $n$ ?

pbornsztein

Is the challenging one an open question? Or do you have the answer (and then we have to move this post into the 'proposed and own problems' section?

Pierre.

**Posted:** Mon Jul 26, 2004 2:31 am

thabit

**Posted:** Mon Jul 26, 2004 3:16 am

rgep · Poincare Conjecture Offline Joined: 03 Jan 2005 Posts: 169

**Posted:** Tue Aug 23, 2005 12:59 pm