When can 10101...1 be prime?
We have,
1 is not prime.
101=1+100 is prime.
10101=1+100+100² is not prime. [ Since, 3|10101]
Now,
Let, N= 10101...1 and numbers of zero in N is n(say).
Then N=1+100+100²+...+100ⁿ .........(i)
=(100ⁿ⁺¹-1)/(100-1)
={(10ⁿ⁺¹)²-1}/99
Therefore, 99=(10ⁿ⁺¹-1)(10ⁿ⁺¹+1)/N
If possible, let N is prime then N|10ⁿ⁺¹-1 or, N|10ⁿ⁺¹+1
Now, N=1+100+100²+...+10²ⁿ [By (i)]
>1+10²ⁿ
>1+10ⁿ⁺¹ [Since, 2n>n+1 ∀n≥2]
Then it's can't happen that, N|10ⁿ⁺¹-1 or, N|10ⁿ⁺¹+1
Therefore our assumption N is prime is wrong.
So, N can't be prime because n>1.
Here only one possibility for N is prime which is if n=1. [Where n is the number/s of zero in N]
1 is not prime.
101=1+100 is prime.
10101=1+100+100² is not prime. [ Since, 3|10101]
Now,
Let, N= 10101...1 and numbers of zero in N is n(say).
Then N=1+100+100²+...+100ⁿ .........(i)
=(100ⁿ⁺¹-1)/(100-1)
={(10ⁿ⁺¹)²-1}/99
Therefore, 99=(10ⁿ⁺¹-1)(10ⁿ⁺¹+1)/N
If possible, let N is prime then N|10ⁿ⁺¹-1 or, N|10ⁿ⁺¹+1
Now, N=1+100+100²+...+10²ⁿ [By (i)]
>1+10²ⁿ
>1+10ⁿ⁺¹ [Since, 2n>n+1 ∀n≥2]
Then it's can't happen that, N|10ⁿ⁺¹-1 or, N|10ⁿ⁺¹+1
Therefore our assumption N is prime is wrong.
So, N can't be prime because n>1.
Here only one possibility for N is prime which is if n=1. [Where n is the number/s of zero in N]
