The numbers of non-primes in S

[Albert1]

$S=({10^1+1,10^2+1,---------,10^{1000}+1})$

please prove the non-prime numbers in $S \geq 990$

 

[Albert1]

hint :

Spoiler

determine the condition of n , where those numbers of $"10^n+1"$ are prime
and $1\leq n \leq 1000$

 

[kaliprasad]

hint :

Spoiler

determine the condition of n , where those numbers of $"10^n+1"$ are prime
and $1\leq n \leq 1000$

Spoiler

n cannot be odd > 1 becuase in that case $10^n = - 1$ mod 11 or $10^n+1$ mod 11 = 0

n cannot be a muliple of odd because if it is of odd p then $10^n+1$ is divisible by $10^p + 1$

so possible set of n is 1, and power of 2 that is $2 , 4,8,16, 32,64,128,256,512$ and not necessarily each of them is prime
so maximum prime numbers = 9 and minimum non prime is 991