In short; 20 is of form 3k+2. The set of numbers given (all of form 3k except one which is of form 3k+1) can only add up to numbers of form 3k or 3k+1; they cannot add up to numbers of form 3k+2 (like 20).
The value 30340 is a value of the form 3k+1, so I can't rule that one out. I can rule out 30341, which is of form 3k+2.
Also, it doesn't matter how many calculations I do to find the answer - what matters is how the number of calculations *scales* with an increasing number of elements in the set. If I find a way to solve the set sum problem with ten billion calculations for a set of _any_ size, then that would prove P=NP (since the number of calculations would then be independent of N, therefore polynomial in N)
The overall manuscript hinges on being able to show there exists a set of subset sums, that is exponentially large, where each requires a dedicated calculation.
Which is why dedicated calculations are within the topic I was addressing.
I would be happy to show that such a calculation cannot reduce the number of subset sums needed to be determined from exponential to polynomial. I don't think its something I would include in the manuscript, as it follows quite naturally and obviously from what is already there. But it is a very sound point, and I appreciate yourself making it.
I wrote yourself earlier, and hope you'll consider reading the actual proofs. And trying to understand the approach.
I hope again your day goes great.
