Efficiently computable, nonempty sets with no known element
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I'm looking for sets $S$ of natural numbers with the following properties:
- There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial in the number of digits of $n,$ with reasonable degree/coefficients)
- It is known that $S$ has a small element (say below $10^{1,000}$)
- No element of $S$ has been identified (or is likely to be found with current technology)
- (Optional) $S$ is a singleton
One example would be the singleton $S$ containing the smaller prime factor of Reble's semiprime (a 1,084 digit number which was nonconstructively proven to be semiprime, see http://www.graysage.com/djr/isp.txt). This number has between 350 and 550 digits, and would be easy to verify if found.
What are some other examples of such sets?
computational-mathematics
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
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add a comment |
$begingroup$
I'm looking for sets $S$ of natural numbers with the following properties:
- There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial in the number of digits of $n,$ with reasonable degree/coefficients)
- It is known that $S$ has a small element (say below $10^{1,000}$)
- No element of $S$ has been identified (or is likely to be found with current technology)
- (Optional) $S$ is a singleton
One example would be the singleton $S$ containing the smaller prime factor of Reble's semiprime (a 1,084 digit number which was nonconstructively proven to be semiprime, see http://www.graysage.com/djr/isp.txt). This number has between 350 and 550 digits, and would be easy to verify if found.
What are some other examples of such sets?
computational-mathematics
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
$endgroup$
$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
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– Ian
Dec 7 '18 at 1:42
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44
add a comment |
$begingroup$
I'm looking for sets $S$ of natural numbers with the following properties:
- There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial in the number of digits of $n,$ with reasonable degree/coefficients)
- It is known that $S$ has a small element (say below $10^{1,000}$)
- No element of $S$ has been identified (or is likely to be found with current technology)
- (Optional) $S$ is a singleton
One example would be the singleton $S$ containing the smaller prime factor of Reble's semiprime (a 1,084 digit number which was nonconstructively proven to be semiprime, see http://www.graysage.com/djr/isp.txt). This number has between 350 and 550 digits, and would be easy to verify if found.
What are some other examples of such sets?
computational-mathematics
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
$endgroup$
I'm looking for sets $S$ of natural numbers with the following properties:
- There is a known efficient algorithm for determining whether a given $n$ is in $S$ (let's say "efficient" means polynomial in the number of digits of $n,$ with reasonable degree/coefficients)
- It is known that $S$ has a small element (say below $10^{1,000}$)
- No element of $S$ has been identified (or is likely to be found with current technology)
- (Optional) $S$ is a singleton
One example would be the singleton $S$ containing the smaller prime factor of Reble's semiprime (a 1,084 digit number which was nonconstructively proven to be semiprime, see http://www.graysage.com/djr/isp.txt). This number has between 350 and 550 digits, and would be easy to verify if found.
What are some other examples of such sets?
computational-mathematics
computational-mathematics
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
edited Dec 7 '18 at 1:37
Elliot Glazer
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
asked Dec 7 '18 at 1:29
Elliot GlazerElliot Glazer
528139
528139
$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
$endgroup$
– Ian
Dec 7 '18 at 1:42
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44
add a comment |
$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
$endgroup$
– Ian
Dec 7 '18 at 1:42
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44
$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
$endgroup$
– Ian
Dec 7 '18 at 1:42
$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
$endgroup$
– Ian
Dec 7 '18 at 1:42
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44
add a comment |
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$begingroup$
Aren't these basically just possible outputs of an NP algorithm that isn't also P?
$endgroup$
– Ian
Dec 7 '18 at 1:42
$begingroup$
Well the main difficulty there is ensuring the set is nonempty. Maybe there's a well-known example that I'm unaware of.
$endgroup$
– Elliot Glazer
Dec 7 '18 at 1:44