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The problem I have is bit vague, so I will try to explain as best as I can.

Suppose we have a base number '$x$' and a set of unique number $'a_1', 'a_2', 'a_3', 'a_4'$ and so on

Now $a_1$ perform some operation on $x$ and results in $y_1$ From $y_1$, I should be able to identify that $a_1$ has performed operations on this number

Now on $y_1, a_2$ performed similar operations and results in $y_2$. From $y_2$, I should be able to infer that either $a_1$ or $a_2$ has performed operations on it and none other numbers had ($a_3, a_4$ etc)

So I am looking for some logic, that can help me create this scenario.

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    Any operations can be done. division, multiplication, complex, hashcode anything as long as the result again leads to a unique number.2012-08-26

3 Answers 3

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Assuming the numbers $a_i$ are to be natural numbers, then a simple approach is to define the operation $\otimes:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ such that $a \otimes b=a p(a)^{b}$, where $p(a)$ is the smallest prime that does not divide $a$. Starting with $x=1$ produces $ (((x\otimes a_1)\otimes a_2)\otimes a_3)...=2^{a_1} 3^{a_2} 5^{a_3} ..., $ from which the sequence $(a_1,a_2,a_3,...)$ is easily retrieved.

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The operations need to be specified, otherwise there is no solution. Let me define a family of operations $f_z(x,y)=z$ (one for each real $z$, independent of $x, y$). Clearly you cannot recover $x$ from the results of these operations.

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    So you are asking for an operation. More than one only makes it harder. If somebody else picks the $a$'s you can't guarantee that the results will be different. As you say, if the $a$'s are all primes, multiplication works well (but watch out if $x$ matches one of the $a$'s). My example of having the $a$'s powers of $2$ and the operation addition also works. There are many other possibilities.2012-08-26
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The logic should be similar to Gödel Numbering and the more sophisticated Gödel Numbering for Sequences.