After sieving, the question is:

**"Is there an infinite number of teta-twin"**(teta-twin are just 2 successive numbers, 1 odd and 1 even, or vice-versa).

First pass, you sieve all teta-multiples of $1\cdot 2^0$:

$1\cdot 2^1$, $1\cdot 2^2$, $1\cdot 2^3$, ...

In the new infinite set, there are still an infinite numbers of teta-twins.

second pass, you sieve all teta-multiples of $3\cdot 2^0$:

$3\cdot 2^1$, $3\cdot 2^2$, $3\cdot 2^3$, ...

In the new infinite set, there are still an infinite numbers of teta-twins.

Third pass, you sieve all teta-multiples of $5\cdot 2^0$:

$5\cdot 2^1$, $5\cdot 2^2$, $5\cdot 2^3$, ...

In the new infinite set, there are still an infinite numbers of teta-twins.

After $n$ passes, you sieve all teta-multiples of $(2n-1)\cdot 2^0$:

$(2n-1)\cdot 2^1$, $(2n-1)\cdot 2^2$, $(2n-1)\cdot 2^3$, ...

In the new infinite set, there are still an infinite numbers of teta-twins.

Your answer: There are infinitely many teta-twins.

My answer:

**There are none**.