Is it possible to find a proper divisor of a composite Fermat Number without using any arithmetic operation...












0












$begingroup$


I have an absurdly simple algorithm which for N = 2^{2^{k-1}} +1 a composite Fermat Number,
computes a divisor 1


The algorithm is also a test for primality, i.e. N is prime if and only if my algorithm
has output e=N.



Features:




  1. The algorithm is a k-step recursion. Hence it contains no steps which involve "Trial & Error".


  2. The algorithm involves simple manipulations of bit strings. It uses non of the
    arithmetic operations +,-,*,/.



My goal is to prove that no such algorithm as I have just described is possible. This would be the contradiction in a theorem I proving.










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$endgroup$








  • 2




    $begingroup$
    All of those arithmetic operations can be implemented as simple manipulations of bit strings.
    $endgroup$
    – user3482749
    Nov 24 '18 at 17:36










  • $begingroup$
    Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
    $endgroup$
    – JoshuaZ
    Nov 24 '18 at 17:57










  • $begingroup$
    Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:25












  • $begingroup$
    What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:28
















0












$begingroup$


I have an absurdly simple algorithm which for N = 2^{2^{k-1}} +1 a composite Fermat Number,
computes a divisor 1


The algorithm is also a test for primality, i.e. N is prime if and only if my algorithm
has output e=N.



Features:




  1. The algorithm is a k-step recursion. Hence it contains no steps which involve "Trial & Error".


  2. The algorithm involves simple manipulations of bit strings. It uses non of the
    arithmetic operations +,-,*,/.



My goal is to prove that no such algorithm as I have just described is possible. This would be the contradiction in a theorem I proving.










share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    All of those arithmetic operations can be implemented as simple manipulations of bit strings.
    $endgroup$
    – user3482749
    Nov 24 '18 at 17:36










  • $begingroup$
    Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
    $endgroup$
    – JoshuaZ
    Nov 24 '18 at 17:57










  • $begingroup$
    Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:25












  • $begingroup$
    What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:28














0












0








0





$begingroup$


I have an absurdly simple algorithm which for N = 2^{2^{k-1}} +1 a composite Fermat Number,
computes a divisor 1


The algorithm is also a test for primality, i.e. N is prime if and only if my algorithm
has output e=N.



Features:




  1. The algorithm is a k-step recursion. Hence it contains no steps which involve "Trial & Error".


  2. The algorithm involves simple manipulations of bit strings. It uses non of the
    arithmetic operations +,-,*,/.



My goal is to prove that no such algorithm as I have just described is possible. This would be the contradiction in a theorem I proving.










share|cite|improve this question









$endgroup$




I have an absurdly simple algorithm which for N = 2^{2^{k-1}} +1 a composite Fermat Number,
computes a divisor 1


The algorithm is also a test for primality, i.e. N is prime if and only if my algorithm
has output e=N.



Features:




  1. The algorithm is a k-step recursion. Hence it contains no steps which involve "Trial & Error".


  2. The algorithm involves simple manipulations of bit strings. It uses non of the
    arithmetic operations +,-,*,/.



My goal is to prove that no such algorithm as I have just described is possible. This would be the contradiction in a theorem I proving.







number-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 24 '18 at 16:58









Arthur LubocceArthur Lubocce

1




1








  • 2




    $begingroup$
    All of those arithmetic operations can be implemented as simple manipulations of bit strings.
    $endgroup$
    – user3482749
    Nov 24 '18 at 17:36










  • $begingroup$
    Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
    $endgroup$
    – JoshuaZ
    Nov 24 '18 at 17:57










  • $begingroup$
    Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:25












  • $begingroup$
    What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:28














  • 2




    $begingroup$
    All of those arithmetic operations can be implemented as simple manipulations of bit strings.
    $endgroup$
    – user3482749
    Nov 24 '18 at 17:36










  • $begingroup$
    Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
    $endgroup$
    – JoshuaZ
    Nov 24 '18 at 17:57










  • $begingroup$
    Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:25












  • $begingroup$
    What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
    $endgroup$
    – Arthur Lubocce
    Nov 26 '18 at 5:28








2




2




$begingroup$
All of those arithmetic operations can be implemented as simple manipulations of bit strings.
$endgroup$
– user3482749
Nov 24 '18 at 17:36




$begingroup$
All of those arithmetic operations can be implemented as simple manipulations of bit strings.
$endgroup$
– user3482749
Nov 24 '18 at 17:36












$begingroup$
Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
$endgroup$
– JoshuaZ
Nov 24 '18 at 17:57




$begingroup$
Proving in general that an algorithm doesn't exist is extremely difficult. For example, the following question is open: Is there are a linear time algorithm for multiplying two positive integers?
$endgroup$
– JoshuaZ
Nov 24 '18 at 17:57












$begingroup$
Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
$endgroup$
– Arthur Lubocce
Nov 26 '18 at 5:25






$begingroup$
Sure, arithmetic operations can be implemented as simple manipulations on bit strings. But with my algorithm the point is that none of what I do amounts to performing any arithmetic operations.
$endgroup$
– Arthur Lubocce
Nov 26 '18 at 5:25














$begingroup$
What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
$endgroup$
– Arthur Lubocce
Nov 26 '18 at 5:28




$begingroup$
What I am attempting to show is the opposite of finding some optimal algorithm. I seek to prove that all algorithms to find divisors of Fermat Numbers or to prove they are composite requires some minimal operations, namely arithmetic operations.
$endgroup$
– Arthur Lubocce
Nov 26 '18 at 5:28










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