This question comes from some reasoning I made myself about a "joke block chain" where every new block is labeled with a triplet **<S, P, N>** where where *S = sum of the N transactions so far* and *P = product of the N transactions*.


So let's say we start with:



```
<5, 5, 1> => [5]
<- include a transaction of 3
<8, 15, 2> => [5, 3] or [3, 5]
<- merge with another branch having <5, 6, 2> ([3,2] or [2,3])
<13, 90, 4> => [5, 3, 3, 2] <- in any order
...

```

I hypothesise that such triplets are unique because it seems to me that *N* limits the possible groupings of *P* factors and *S* identifies one of those gropings, and you can't apply tricks like "keep multiplying by 1 to tune the sum" because then *N* will also change.


Is there any literature about this? How could I demonstrate the uniqueness of such triplets?


Given one of those triplets, can the problem of identifying the "original" numbers be considered hard? I know factorisation is NP, does adding *S* and *N* change its complexity?


Extra question: could there be a way to check if a triplet is "valid" (meaning there actually exists a set of *N* integers that satisfy the *S* and *P* constraints) without actually solving it?



Are <sum, product, N> triplets unique and hard to solve?
If I understand the question, then the triple $(30,840,3)$ could come from $6+10+14=30$, $6\times10\times14=840$ or from $7+8+15=30$, $7\times8\times15=840$.


