I am working on a project where I am trying to achieve a secure way to transmit data as numbers. It aims to find the maximum number that could not be derived from addition of given three or more numbers.
For example; $A = \{3,7,9\}$
This problem can be formulated as:
$3a+7b+9c\ne x$ where $a,b,c\ge0$
and then finding the maximum by hand.
$x$ can be equal to $3,6,7,9,10,12,13,14,15,16,17,...$
Therefore, for $A$ the answer is 11 as all the positive real numbers bigger than 11 can be written as a summation of 3, 7 and 9.
Is there anyway I could find the numbers that apply to the rule for a given set without calculating it manually?
I found a code to Find the smallest positive integer value that cannot be represented as sum of any subset of a given array but nothing about biggest positive integer.
Your problem is equivalent to finding the frobenius number of a numerical semigroup with generators $\{3,7,9\}$ which is equivalent to the semigroup generated by $\{3,7\}$ since $9$ can generated by $3\cdot 3$.
Usually, finding the frobenius number is non-trivial, but there is an explicit formula if the semigroup is generated by just two generators $x$ and $y$: $xy-x-y$. In the case of $x=3$ and $y=7$, we have $xy-x-y = 3(7)-3-7 =11$ which is consistent with your result.
Resources: https://en.m.wikipedia.org/wiki/Coin_problem https://en.m.wikipedia.org/wiki/Numerical_semigroup
Edit: if you'd like to leave the problem for a CAS to solve, then I know GAP has some methods for calculating things involving numerical semigroups, possibly including the frobenius number. There is also a Sage Numerical Semigroups wrapper out there if you'd prefer to call those GAP functions from within Sage.