## ‘Reversing’ the carry operation in multiplication [closed]

Hello I'm not a mathematician so forgive me if I'm being unclear. I would like to know if there is a way to apply inverse 'carry' operations on the product, A, of two unknown numbers, written in any base, to find the 'number', B, (or rather the polynomial in p which is the same number when evaluated at $p=base$, see 1) which would be the result of the product without applying the carry operations.

$(1) \sum_{k=0}^n a_kp^k = \sum_{k=0}^m b_kp^k = number, n \ge m$

Example in base two: $1001 \to 121$ since without carry $11*11=121$.

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Any question that begins "Hello I'm not a mathematician" is going to be closed. – tergi Aug 30 at 17:59
No, because two different pairs of numbers, whose products as polynomials in $p$ are two a different polynomials, can have the same product, e.g. $3\times 3=9\times 1$ in base $2$. Anyways, this is in no way a research-level math question and mathoverflow.net is not an appropriate site for it. – Will Sawin Aug 30 at 18:03
From the FAQ: "MathOverflow's primary goal is for users to ask and answer research level math questions.... If your question is closed as 'off topic,' it might be because it was too elementary." (Tergi's comment gives me a strange urge to begin my next question with "Hello I'm not a mathematician" just to see what happens.) – Trevor Wilson Aug 30 at 18:04
@Trevor: but you play one on the internet. :-) – Todd Trimble Aug 30 at 18:13

## closed as off topic by Henry Cohn, Bill Johnson, Will Sawin, Noah Stein, Emil JeřábekAug 30 at 18:05

Your "inverse" operation is not clearly defined. You need some condition on the $b_k$ in order to ensure uniqueness. For instance, $1001$, $121$, $113$ all evaluate to the same value in base two.