When using as the set of coins all logarithms of the prime numbers or numbers in general, and when using the logarithm of the number to be factored. The problem is just finding the logarithms that can be used to make the total. But then the change making problem is listed as non NP problem, but the prime factorization problem is found to be NP complete. Can anyone explain why this is?
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When using as the set of coins all logarithms of the prime numbers or numbers in general
The change-making typically assumes a finite set of coins.
You can, of course, extend it to infinite sets of coins but then you get a different problem. Depending on the exact formulation, it seems plausible that integer factorization would reduce to that new problem.
But then the change making problem is listed as non NP problem, but the prime factorization problem is found to be NP complete.
Careful with your terminology there.
The general change-making problem is certainly in NP since it can be solved by integer programming. I do not know if it is NP-hard, but the common algorithms do not show that it is in P since they are only pseudo-polynomial. This shows, at least, that the problem is not strongly NP-hard.
Real-world change-making usually deals with canonical sets of coins for which a simple greedy algorithm works; it runs in constant time assuming a constant set of coins and the uniform cost model.
Integer factoring, on the other hand, is not known to be in P nor NP-hard; it's an open problem, and if P ≠ NP it may be in neither (but in NPI instead).
