Manipulating sets into periodic sets

Question

Consider the set $[0,1,1,0,1,0,1,0,0,0,1,...]$. The nth element in this set is equal to $1$ if n is prime and is else equal to $0$. Here are my question:

1. Is this Set non-periodic (If you write it as the fractional part of a binary number, the number is irrational)? To me it seems pretty obvious, because if it would be periodic we would have formula for the nth prime, but i cant come up with a explicite proof

2. If we turn an infinite amount of $1$'s into $0$'s so that the set still contains of an infinite amount of $1$'s, can we create an periodic set? I think the answer is no, but i cannot come up with a proof of this either

Answer 1

For question 1, since prime gaps are unbounded, your sequence of $0$s and $1$s is non-periodic.

Answer 2

1) Suppose that the period is $k$. Now, consider the $k$ composite numbers $(k+1)!+2,(k+1)!+3,\ldots,(k+1)!+(k+1)$. Their existence and the period implies that all elements of the sequence must be $1$, which clearly isn’t the case. Contradiction.

2) Suppose that after turning $1$s into $0$s, the number $n$ is still a $1$, and the period of the new sequence is $k$. Then, the number $nk+n$, which isn’t prime, should also be a $1$, which is absurd. Again, contradiction.