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Is there a solution for $2^{n-1}equiv 2^{16}+1mod n$ or $2^{n-1}equiv 2^{26}+1mod n$?

1

1

$begingroup$

Related to this question : Can I find all solutions of $2^{n-1}equiv kmod n$?

Does one of the congruences $$2^{n-1}equiv 2^{16}+1mod n$$ and $$2^{n-1}equiv 2^{26}+1mod n$$ have an integer solution $n>1$ ?

Enzo Creti checked the second congruences upto $ 127cdot 10^9 $. No solution was found.

The first congruence has no solution below $10^9$

number-theory elementary-number-theory modular-arithmetic

share|cite|improve this question

edited Mar 22 at 13:01

Peter

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $$n = 4 428 169 422 323$$ solves $$2^{n-1}equiv 2^{16}+1mod n$$
  $endgroup$
  – Peter
  Mar 27 at 18:48
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Related to this question : Can I find all solutions of $2^{n-1}equiv kmod n$?

Does one of the congruences $$2^{n-1}equiv 2^{16}+1mod n$$ and $$2^{n-1}equiv 2^{26}+1mod n$$ have an integer solution $n>1$ ?

Enzo Creti checked the second congruences upto $ 127cdot 10^9 $. No solution was found.

The first congruence has no solution below $10^9$

number-theory elementary-number-theory modular-arithmetic

share|cite|improve this question

edited Mar 22 at 13:01

Peter

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $$n = 4 428 169 422 323$$ solves $$2^{n-1}equiv 2^{16}+1mod n$$
  $endgroup$
  – Peter
  Mar 27 at 18:48
  

  
  
  
  

add a comment | 

$begingroup$

Related to this question : Can I find all solutions of $2^{n-1}equiv kmod n$?

Does one of the congruences $$2^{n-1}equiv 2^{16}+1mod n$$ and $$2^{n-1}equiv 2^{26}+1mod n$$ have an integer solution $n>1$ ?

Enzo Creti checked the second congruences upto $ 127cdot 10^9 $. No solution was found.

The first congruence has no solution below $10^9$

number-theory elementary-number-theory modular-arithmetic

share|cite|improve this question

edited Mar 22 at 13:01

Peter

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

$endgroup$

share|cite|improve this question

edited Mar 22 at 13:01

Peter

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

share|cite|improve this question

edited Mar 22 at 13:01

Peter

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

asked Mar 22 at 12:47

PeterPeter

49.1k1240138

asked Mar 22 at 12:47

PeterPeter

49.1k1240138