Prove that 10101...10101 is NOT a prime.

Solution 1:

Note that $1010101....10101$ is $\frac{10^{4034}-1}{99}$.

Also, $10^{4034}-1$ is $(10^{2017}-1)(10^{2017}+1)$, both of which are larger than $99$.

This implies that the number is not prime.

Solution 2:

And since you have asked this question, here is an interesting piece of information, that in the sequence $101,10101,1010101,....$ none of the numbers are prime EXCEPT the first one. This can be proved quite easily and in your case the number is $${{(10^{4034}-1)}/99}$$, and the numerator can be written as $${{(10^{2017}}-1)} \times {{(10^{2017}}+1)}$$ and the first multiplicand has a factor 10-1=9 and the second multiplicand has a factor 10+1=11 so their product is divisible by 9 and 11, which are coprime,hence divisible by 99, so the number is not prime, because it has 2 factors now both greater than 1. So it is a prime which is anyway supported by the result above.