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Find all positive integers n < 10^100 which simultaneously n divides 2^n, n - 1 divides 2^n - 1, and n -2 divides 2^n - 2. Thank you for any help.

 Jun 23, 2019
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n=3; a=if(2^n%n==0 and (2^n-1)%(n-1)==0 and (2^n-2)%(n-2)==0, goto2, goto3);printa,n;n++;if(n<100000, goto1, discard=0;

 

n =4,  16,  65,536, (2^65,536) - This last number is way > 10^100.


So, there are only 3 such numbers and they all satisfy the conditions in the question. Example:2^16 =65,536.
65,536 mod 16=0. 65,536 - 1 mod (16 - 1) = 0. 65,536 -2 mod(16 - 2) =0.

 Jun 23, 2019
edited by Guest  Jun 23, 2019

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