Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible.

However, is it possible to divide each interval $R_i$ into $f(k,t)$ non-overlapping intervals for some polynomial $f$ such that the total number of integer points in $S_i$, the $\underline{\mbox{union of a subset}}$ of the non-overlapping intervals, satisfies $|S_i|>{2^{t-\delta(t,k)}}$ for some function $\delta(t,k)$ whose range on positive $t,k$ is bounded below by $0$ such that if there is a solution to $\sum_{i=1}^k a_ix_i = N$ with $x_i\in S_i$ then the solution is unique? What is the function $\delta(t,k)$ and how to find the solution $x_i\in S_i$?

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    $\begingroup$ What happens in the case $k=2$? And what do you do if there is no solution with $x_i\lt a_i$? $\endgroup$ – Gerry Myerson Sep 14 '13 at 23:03
  • $\begingroup$ @GerryMyerson I am just seeing if it is possible to design some parameters. It seems to come down to the case whether I can force such a linear system to have an unique solution. I have control over $a_i$s and I know that my system always has one solution. However the design forces me to have only one solution and I looking for the possibility that if I constrain my ranges for $x_i$s, may be I can force it to have always one solution. Even for $k=2$, I know that I will have a solution. $\endgroup$ – Brout Sep 14 '13 at 23:55
  • $\begingroup$ May be should I ask it this way? ".... is it possible to find regions such that if the system has one solution it is unique". $\endgroup$ – Brout Sep 15 '13 at 0:04
  • $\begingroup$ I changed the question. May be this is more correct. $\endgroup$ – Brout Sep 15 '13 at 0:05

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