$2^i - 2293$ is always composite?












12














Is $2^i - 2293$ always composite for $i=1,2,3,...$ ?



I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$




In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}]



Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943,
1}}, {{6737807, 1}, {83550917, 1}}, {{399550573,
1}, {23638391743063, 1}}, {{281, 1}, {14821, 1}, {24203,
1}, {3712421, 1}, {423447263633, 1}}, {{149, 1}, {9492181,
1}, {1879650895890301462105483811, 1}}, {{137, 1}, {2683,
1}, {2360851, 1}, {2808601, 1}, {2020240309, 1}, {9058295304389951,
1}}, {{23, 1}, {29, 1}, {107, 1}, {199, 1}, {21035159,
1}, {5797034797, 1}, {28376991193, 1}, {15226094729816791,
1}}, {{526557780757, 1}, {1946642765756893,
1}, {12247765663995514289321022531499, 1}}, {{47, 1}, {617,
1}, {160191103, 1}, {8207681257, 1}, {9477520181923,
1}, {46405673331331, 1}, {12560339159195827, 1}}, {{439,
1}, {80494171516099513876232232380087403910135940632146649572738323
52130381, 1}}}






number-theory prime-factorization







share|cite|improve this question




















  • 3




    $$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
    – Peter
    Nov 18 '15 at 19:10








  • 1




    I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
    – Peter
    Nov 19 '15 at 18:54








  • 2




    It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
    – Mirko
    Nov 20 '15 at 14:08






  • 4




    I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
    – Peter
    Nov 20 '15 at 19:13








  • 3




    @a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
    – Gottfried Helms
    Nov 22 '15 at 7:37


















12














Is $2^i - 2293$ always composite for $i=1,2,3,...$ ?



I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$




In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}]



Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943,
1}}, {{6737807, 1}, {83550917, 1}}, {{399550573,
1}, {23638391743063, 1}}, {{281, 1}, {14821, 1}, {24203,
1}, {3712421, 1}, {423447263633, 1}}, {{149, 1}, {9492181,
1}, {1879650895890301462105483811, 1}}, {{137, 1}, {2683,
1}, {2360851, 1}, {2808601, 1}, {2020240309, 1}, {9058295304389951,
1}}, {{23, 1}, {29, 1}, {107, 1}, {199, 1}, {21035159,
1}, {5797034797, 1}, {28376991193, 1}, {15226094729816791,
1}}, {{526557780757, 1}, {1946642765756893,
1}, {12247765663995514289321022531499, 1}}, {{47, 1}, {617,
1}, {160191103, 1}, {8207681257, 1}, {9477520181923,
1}, {46405673331331, 1}, {12560339159195827, 1}}, {{439,
1}, {80494171516099513876232232380087403910135940632146649572738323
52130381, 1}}}






number-theory prime-factorization







share|cite|improve this question




















  • 3




    $$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
    – Peter
    Nov 18 '15 at 19:10








  • 1




    I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
    – Peter
    Nov 19 '15 at 18:54








  • 2




    It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
    – Mirko
    Nov 20 '15 at 14:08






  • 4




    I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
    – Peter
    Nov 20 '15 at 19:13








  • 3




    @a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
    – Gottfried Helms
    Nov 22 '15 at 7:37
















12












12








12


6





Is $2^i - 2293$ always composite for $i=1,2,3,...$ ?



I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$




In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}]



Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943,
1}}, {{6737807, 1}, {83550917, 1}}, {{399550573,
1}, {23638391743063, 1}}, {{281, 1}, {14821, 1}, {24203,
1}, {3712421, 1}, {423447263633, 1}}, {{149, 1}, {9492181,
1}, {1879650895890301462105483811, 1}}, {{137, 1}, {2683,
1}, {2360851, 1}, {2808601, 1}, {2020240309, 1}, {9058295304389951,
1}}, {{23, 1}, {29, 1}, {107, 1}, {199, 1}, {21035159,
1}, {5797034797, 1}, {28376991193, 1}, {15226094729816791,
1}}, {{526557780757, 1}, {1946642765756893,
1}, {12247765663995514289321022531499, 1}}, {{47, 1}, {617,
1}, {160191103, 1}, {8207681257, 1}, {9477520181923,
1}, {46405673331331, 1}, {12560339159195827, 1}}, {{439,
1}, {80494171516099513876232232380087403910135940632146649572738323
52130381, 1}}}






number-theory prime-factorization







share|cite|improve this question















Is $2^i - 2293$ always composite for $i=1,2,3,...$ ?



I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$




In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}]



Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943,
1}}, {{6737807, 1}, {83550917, 1}}, {{399550573,
1}, {23638391743063, 1}}, {{281, 1}, {14821, 1}, {24203,
1}, {3712421, 1}, {423447263633, 1}}, {{149, 1}, {9492181,
1}, {1879650895890301462105483811, 1}}, {{137, 1}, {2683,
1}, {2360851, 1}, {2808601, 1}, {2020240309, 1}, {9058295304389951,
1}}, {{23, 1}, {29, 1}, {107, 1}, {199, 1}, {21035159,
1}, {5797034797, 1}, {28376991193, 1}, {15226094729816791,
1}}, {{526557780757, 1}, {1946642765756893,
1}, {12247765663995514289321022531499, 1}}, {{47, 1}, {617,
1}, {160191103, 1}, {8207681257, 1}, {9477520181923,
1}, {46405673331331, 1}, {12560339159195827, 1}}, {{439,
1}, {80494171516099513876232232380087403910135940632146649572738323
52130381, 1}}}






number-theory prime-factorization




number-theory prime-factorization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 22 '18 at 12:34









amWhy

192k28225439




192k28225439










asked Nov 15 '15 at 7:57









a boy

422211




422211








  • 3




    $$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
    – Peter
    Nov 18 '15 at 19:10








  • 1




    I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
    – Peter
    Nov 19 '15 at 18:54








  • 2




    It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
    – Mirko
    Nov 20 '15 at 14:08






  • 4




    I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
    – Peter
    Nov 20 '15 at 19:13








  • 3




    @a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
    – Gottfried Helms
    Nov 22 '15 at 7:37
















  • 3




    $$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
    – Peter
    Nov 18 '15 at 19:10








  • 1




    I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
    – Peter
    Nov 19 '15 at 18:54








  • 2




    It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
    – Mirko
    Nov 20 '15 at 14:08






  • 4




    I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
    – Peter
    Nov 20 '15 at 19:13








  • 3




    @a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
    – Gottfried Helms
    Nov 22 '15 at 7:37










3




3




$$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
– Peter
Nov 18 '15 at 19:10






$$|2^n-2293|$$ is composite for $1le nle 200,000$, so a prime of the desired form must have more than $60,000$ digits.
– Peter
Nov 18 '15 at 19:10






1




1




I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
– Peter
Nov 19 '15 at 18:54






I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :)
– Peter
Nov 19 '15 at 18:54






2




2




It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
– Mirko
Nov 20 '15 at 14:08




It appears that $2293$ is interesting for another, related reason: primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $kcdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238
– Mirko
Nov 20 '15 at 14:08




4




4




I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
– Peter
Nov 20 '15 at 19:13






I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits!
– Peter
Nov 20 '15 at 19:13






3




3




@a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
– Gottfried Helms
Nov 22 '15 at 7:37






@a boy, please replace your commercialized link (at adf.ly) by the direct link math.stackexchange.com/questions/597234/… . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually
– Gottfried Helms
Nov 22 '15 at 7:37












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