prove there are no more than 9 primes whose decimal representation is a string of ones

Question

paxpolaris · Answer

Any hints on how to solve this....

All I got so far...
If the length of the string is divisible by 3, then the number is divisible by 3
If the length of the string is even, then the number is divisible by 11

anonymous · Answer

Hang on, I forgot - above 10 and below 10^29....

paxpolaris · Answer

So strings of length 2 to 29... that's 28 numbers
11 is prime...
   Using the rules above we can eliminate strings of length:
3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28

   We are left with:
2, 5, 7, 11, 13, 17, 19, 21, 23, 25, 29
that's 11 get rid of 2 more

kinggeorge · Answer

While not the best method, now that we've reduced the possibilities to 11 a brute force attack might work.

kinggeorge · Answer

Although, in that list, there are 9 primes and 2 composite numbers. I might also try to show that strings of length 21 and 25 are composite.

paxpolaris · Answer

sorry 21 is already eliminated ... that's 10

anonymous · Answer

multiples of 5 have 41
multiples of 7 have 239, 4649
11 has multiples of .... never mind
4 has 11, 101, 1111
6 has 7, 13, 21.
Get rid of 3, 5, 7, 11
after some random working out, 
19 and 23 are the only two that work. Thanks.

anonymous · Answer

other than 2...

kinggeorge · Answer

Then we're done. Now that we've eliminated at least 2 from the list, there can be no more than 9 primes whose decimal representation is a string of ones between 10 and $10^{29}$