XOR-Primes (noch nicht übersetzt)

We use $x\oplus y$ for the bitwise XOR of $x$ and $y$.

Define the XOR-product of $x$ and $y$, denoted by $x \otimes y$, similar to a long multiplication in base $2$, except that the intermediate results are XORed instead of the usual integer addition.

For example, $7 \otimes 3 = 9$, or in base $2$, $111_2 \otimes 11_2 = 1001_2$:

$$ \begin{align*} \phantom{\otimes 111} 111_2 \\ \otimes \phantom{1111} 11_2 \\ \hline \phantom{\otimes 111} 111_2 \\ \oplus \phantom{11} 111_2 \phantom{9} \\ \hline \phantom{\otimes 11} 1001_2 \\ \end{align*} $$

An XOR-prime is an integer $n$ greater than $1$ that is not an XOR-product of two integers greater than $1$. The above example shows that $9$ is not an XOR-prime. Similarly, $5 = 3 \otimes 3$ is not an XOR-prime. The first few XOR-primes are $2, 3, 7, 11, 13, ...$ and the 10th XOR-prime is $41$.

Find the $5\,000\,000$th XOR-prime.