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Is this “classic probability view†on Fermat Last Theorem studied somewhere?
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I was wondering where (or maybe even if) is this simple reformulation of Fermat's Last Theorem studied:
There are no two events $A,B$ with classical probabilities $P(A), P(B) in mathbbQ cap (0,1)$ satisfying following criterion: In $n>2$ tries, the probability of $A$ happening exactly $n$ times is the same as probability of $B$ not happening at least once.
If there were such events, and $P(A)=a/b$, $P(B)=c/d$, then we would have $(a/b)^n=1-(c/d)^n$, which means $(ad)^n+(cb)^n=(bd)^n$ and we would have solution to FLT (which we know is not possible of course).
Any literature that is viewing the problem this way would be interesting, similarly to situation where certain algebraic problems can be solved by looking at them from geometric point of view. In this case there could be some interesting line of thought when looking at the problem from probability point of view - what would such two events imply? Could we perhaps construct some another interesting events from these? Could we try to use some basic theorems from probability like Bayes's theorem? ...)
Searched through Fermat's Last Theorem for Amateurs by Ribenboim P., but nothing related seems to be there. Searching on MSE, the $p^n+q^n=1$ equation is here couple times, but usually to be studied from algebraic point of view. Maybe this approach is just so fruitless that no one bothers to mention it? ...
number-theory reference-request conjectures
asked Jul 26 at 20:24
Sil
5,11821342
add a comment |Â
up vote
3
down vote
favorite
I was wondering where (or maybe even if) is this simple reformulation of Fermat's Last Theorem studied:
There are no two events $A,B$ with classical probabilities $P(A), P(B) in mathbbQ cap (0,1)$ satisfying following criterion: In $n>2$ tries, the probability of $A$ happening exactly $n$ times is the same as probability of $B$ not happening at least once.
If there were such events, and $P(A)=a/b$, $P(B)=c/d$, then we would have $(a/b)^n=1-(c/d)^n$, which means $(ad)^n+(cb)^n=(bd)^n$ and we would have solution to FLT (which we know is not possible of course).
Any literature that is viewing the problem this way would be interesting, similarly to situation where certain algebraic problems can be solved by looking at them from geometric point of view. In this case there could be some interesting line of thought when looking at the problem from probability point of view - what would such two events imply? Could we perhaps construct some another interesting events from these? Could we try to use some basic theorems from probability like Bayes's theorem? ...)
Searched through Fermat's Last Theorem for Amateurs by Ribenboim P., but nothing related seems to be there. Searching on MSE, the $p^n+q^n=1$ equation is here couple times, but usually to be studied from algebraic point of view. Maybe this approach is just so fruitless that no one bothers to mention it? ...
number-theory reference-request conjectures
asked Jul 26 at 20:24
Sil
5,11821342
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I was wondering where (or maybe even if) is this simple reformulation of Fermat's Last Theorem studied:
There are no two events $A,B$ with classical probabilities $P(A), P(B) in mathbbQ cap (0,1)$ satisfying following criterion: In $n>2$ tries, the probability of $A$ happening exactly $n$ times is the same as probability of $B$ not happening at least once.
If there were such events, and $P(A)=a/b$, $P(B)=c/d$, then we would have $(a/b)^n=1-(c/d)^n$, which means $(ad)^n+(cb)^n=(bd)^n$ and we would have solution to FLT (which we know is not possible of course).
Any literature that is viewing the problem this way would be interesting, similarly to situation where certain algebraic problems can be solved by looking at them from geometric point of view. In this case there could be some interesting line of thought when looking at the problem from probability point of view - what would such two events imply? Could we perhaps construct some another interesting events from these? Could we try to use some basic theorems from probability like Bayes's theorem? ...)
Searched through Fermat's Last Theorem for Amateurs by Ribenboim P., but nothing related seems to be there. Searching on MSE, the $p^n+q^n=1$ equation is here couple times, but usually to be studied from algebraic point of view. Maybe this approach is just so fruitless that no one bothers to mention it? ...
number-theory reference-request conjectures
asked Jul 26 at 20:24
Sil
5,11821342
I was wondering where (or maybe even if) is this simple reformulation of Fermat's Last Theorem studied:
There are no two events $A,B$ with classical probabilities $P(A), P(B) in mathbbQ cap (0,1)$ satisfying following criterion: In $n>2$ tries, the probability of $A$ happening exactly $n$ times is the same as probability of $B$ not happening at least once.
If there were such events, and $P(A)=a/b$, $P(B)=c/d$, then we would have $(a/b)^n=1-(c/d)^n$, which means $(ad)^n+(cb)^n=(bd)^n$ and we would have solution to FLT (which we know is not possible of course).
Any literature that is viewing the problem this way would be interesting, similarly to situation where certain algebraic problems can be solved by looking at them from geometric point of view. In this case there could be some interesting line of thought when looking at the problem from probability point of view - what would such two events imply? Could we perhaps construct some another interesting events from these? Could we try to use some basic theorems from probability like Bayes's theorem? ...)
Searched through Fermat's Last Theorem for Amateurs by Ribenboim P., but nothing related seems to be there. Searching on MSE, the $p^n+q^n=1$ equation is here couple times, but usually to be studied from algebraic point of view. Maybe this approach is just so fruitless that no one bothers to mention it? ...
number-theory reference-request conjectures
asked Jul 26 at 20:24
Sil
5,11821342
edited Jul 26 at 20:34
asked Jul 26 at 20:24
Sil
5,11821342
asked Jul 26 at 20:24
Sil
5,11821342
asked Jul 26 at 20:24
Sil
5,11821342
5,11821342
add a comment |Â
add a comment |Â
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