Firstly, we see that the number 10001 is not a Prime since it can be expressed as the product of the two numbers 73 and 137. That is,
10001 = 73 * 137
Similarly, we have:
1000100010001 = 73 * 13700000137
........ and so on.
Therefore, we can see that the numbers which contain '2n' number of 1's (Ones), where 'n' is a Positive Integer is always divisible by 73 and therefore cannot be a Prime.
Secondly, we know that the numbers the sum of whose digits is divisible by 3 (or is a multiple of 3) is divisible by 3. So all the numbers of the given sequence(of the given form) containing '3n' number of 1's (Ones), where 'n' is a Positive Integer is obviously divisible by 3 and therefore not a prime.
What we have obtained till now is that the numbers of the given form which contain '2n' and '3n' number of 1's (Ones), that is 2,3,4,6,8,9,10,12,14,15,16,18,20,........ number of Ones) can never be Primes.
Now someone Please help me with the rest of the solution, that is to prove that the number with 5,7,11,13,17,19,........ number of Ones cannot be a Prime.