Project Euler Problem 810
We use xoplus y for the bitwise XOR of x and y.
Solution
Answer: 124136381
The XOR-product is exactly multiplication of binary polynomials over the field $\mathbb{F}_2$ (carryless multiplication). Thus, XOR-primes are precisely the irreducible polynomials over $\mathbb{F}_2$, interpreted as integers via their binary coefficients. This is sequence A014580 in OEIS (“binary irreducible polynomials evaluated at $x=2$”).
Counting irreducible polynomials by degree using the standard formula
$$I(n)=\frac{1}{n}\sum_{d\mid n}\mu(d),2^{n/d}$$
shows that the cumulative number through degree $25$ is $2,807,196$, while through degree $26$ it exceeds $5,000,000$. Therefore, the target is the
$$5,000,000 - 2,807,196 = 2,192,804$$
th irreducible polynomial of degree $26$. A sieve over monic odd degree-26 binary polynomials (eliminating products with irreducible factors of degree $\le 13$) yields the target integer:
$$124136381$$
Answer: 124136381