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The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and to get a whole number (I didn't include $1$ for obvious reasons) and let $G_2=\{6^2,6^3,6^4,6^6\}$ be the numbers that can not be used in $z^{G_1}$. We call $G$ as the union of $G_1$ and $G_2$.

Now see a table:

From OUR point of view:

If F had numbers in the red row he could easily guess S's number. He says I don't know, which means he has $6,6^2,6^3,6^4,6^6$.We are confused between green cells and red cells, for $6^6$ between green and blue cell.

S says at first he didn't know, which means he had not a number in red column, if not he could easily guessed F's number. Also he knows that F does not have a number in red row, if F had he would guess S's number. Since we eliminated red rows now we are not confused in yellow and red cells. The only possible situation that F and S has that conversation is that both of them has $6^6$.

Mission accomplished.

As the OP noted I took $(x,y)$ different than $(y,x)$, if not there would be multiple solutions. To see the solutions color yellow cells to green.

The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and to get a whole number (I didn't include $1$ for obvious reasons) and let $G_2=\{6^2,6^3,6^4,6^6\}$ be the numbers that can not be used in $z^{G_1}$. We call $G$ as the union of $G_1$ and $G_2$.

Now see a table:

From OUR point of view:

If F had numbers in the red row he could easily guess S's number. He says I don't know, which means he has $6,6^2,6^3,6^4,6^6$.We are confused between green cells and red cells, for $6^6$ between green and blue cell.

S says at first he didn't know, which means he didn't have a number in red column, if not he could easily guessed F's number. Also he knows that F does not have a number in red row, if F had he would guess S's number. Since we eliminated red rows now we are not confused in yellow and red cells. The only possible situation that F and S has that conversation is that both of them has $6^6$.

Mission accomplished.

As the OP noted I took $(x,y)$ different than $(y,x)$, if not there would be multiple solutions. To see the solutions color yellow cells to green.

The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and to get a whole number (I didn't include $1$ for obvious reasons) and let $G_2=\{6^2,6^3,6^4,6^6\}$ be the numbers that can not be used in $z^{G_1}$. We call $G$ as the union of $G_1$ and $G_2$.

Now see a table:

From OUR point of view:

If F had numbers in the red row he could easily guess S's number. He says I don't know, which means he has $6,6^2,6^4,6^6$.We are confused between green cells and red cells, for $6^6$ between green and blue cell.

S says at first he didn't know, which means he had not a number in red column, if not he could easily guessed F's number. Also he knows that F does not have a number in red row, if F had he would guess S's number. Since we eliminated red rows now we are not confused in yellow and red cells. The only possible situation that F and S has that conversation is that both of them has $6^6$.

Mission accomplished.

As the OP noted I took $(x,y)$ different than $(y,x)$, if not there would be multiple solutions. To see the solutions color yellow cells to green.

The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and to get a whole number (I didn't include $1$ for obvious reasons) and let $G_2=\{6^2,6^3,6^4,6^6\}$ be the numbers that can not be used in $z^{G_1}$. We call $G$ as the union of $G_1$ and $G_2$.

Now see a table:

From OUR point of view:

If F had numbers in the red row he could easily guess S's number. He says I don't know, which means he has $6,6^2,6^3,6^4,6^6$.We are confused between green cells and red cells, for $6^6$ between green and blue cell.

S says at first he didn't know, which means he had not a number in red column, if not he could easily guessed F's number. Also he knows that F does not have a number in red row, if F had he would guess S's number. Since we eliminated red rows now we are not confused in yellow and red cells. The only possible situation that F and S has that conversation is that both of them has $6^6$.

Mission accomplished.

As the OP noted I took $(x,y)$ different than $(y,x)$, if not there would be multiple solutions. To see the solutions color yellow cells to green.

The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and $6^{12}/z$ to get a whole number (I didn't include $1$ for obvious reasons). The numbers between $12$ and $6^{12}$ can not be used in $6^{12/z}$ because it wouldn't be whole number. We call the set of numbers greater than $12$ as $G_2$.

Now if F has a number $G_1$ then S could have a number in $G_1$ or $G_2$. If F has a number in $G_2$ then S would have a number in $G_2$, then F could easily divide $6^12$ to his number to get S's number. F says I don't know, which means he has a number in $G_1$. S has the same argument, S is also confused.

F says "I don't know your number". S thinks if F does not know my number then he is in my condition. He must have a number in $G_1$. since I know my number I just solve the equation $y^x=6^{12}$ to get $x$. S says "I know your number".

He knows that F has $12$ because he has $6$. These two numbers are the only ones to give $6^{12}$ in $G_1$.

To be better explained (with a table)...

The first mathematician: F

The second mathematician: S

F has $x$ and S has $y$.

F thinks that S has a number in the form of $6^{12/x}$ or $6^{12}/x$.

S thinks that F has a number in the form of $6^{12/y}$ or $6^{12}/y$.

Let's say $G_1=\{2,3,4,6,12\}$ are the sets of number that can be used in $6^{12/z}$ and to get a whole number (I didn't include $1$ for obvious reasons) and let $G_2=\{6^2,6^3,6^4,6^6\}$ be the numbers that can not be used in $z^{G_1}$. We call $G$ as the union of $G_1$ and $G_2$.

Now see a table:

From OUR point of view:

If F had numbers in the red row he could easily guess S's number. He says I don't know, which means he has $6,6^2,6^4,6^6$.We are confused between green cells and red cells, for $6^6$ between green and blue cell.

S says at first he didn't know, which means he had not a number in red column, if not he could easily guessed F's number. Also he knows that F does not have a number in red row, if F had he would guess S's number. Since we eliminated red rows now we are not confused in yellow and red cells. The only possible situation that F and S has that conversation is that both of them has $6^6$.

Mission accomplished.

As the OP noted I took $(x,y)$ different than $(y,x)$, if not there would be multiple solutions. To see the solutions color yellow cells to green.