I have solutions for the following two problems and I am hoping to obtain some feedback on my solutions.
Does the number $0.123456789101112l314\ldots$ which is obtained by writing successively all the integers, represent a rational number?
Does the number $0.011010100010100\ldots$ , where $a_n = 1$ if n is prime, 0 otherwise, represent a rational number?
Solution. My strategy for both problems is to show that there are arbitrarily long chains of zeros and so the decimal representation can not be periodic. Take chain of zeros to mean sandwiched between non-zero integers.
Problem 1. $10^n$ for $n={1,2,\ldots}$ will do
Problem 2. To show that a chain of zeros of length $n-1$ exists, consider $n!+2,n!+3,n!+4,\ldots,n!+(n-1),n!+n$ all of these numbers are composite. Now let $p_0$ be the largest prime less than $n!+2$ and $p_1$ be the smallest prime greater than $n!+2$. Hence, $p_1> n!+n$ and $p_1$ and $p_0$ are consecutive primes. As such, we can create an arbitrarily long chain of 0's.
What do you think?
PS: I didn't know what to say in the title, feel free to edit it