Add solution for Problem 810

project_euler/problem_810/sol1.py

@@ -0,0 +1,158 @@
+"""
+Project Euler Problem 810: https://projecteuler.net/problem=810
+
+We use x ⊕ y for the bitwise XOR of x and y.
+Define the XOR-product of x and y, denoted by x ⊗ y,
+similar to a long multiplication in base 2,
+except the intermediate results are XORed instead of usual integer addition.
+
+For example, 7 ⊗ 3 = 9, or in base 2, 111_2 ⊗ 11_2 = 1001_2:
+        111
+    ⊗   11
+    -------
+        111
+       111
+    -------
+       1001
+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 ⊗ 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.
+
+References:
+http://en.wikipedia.org/wiki/M%C3%B6bius_function
+
+"""
+
+
+def xor_multiply(a: int, b: int) -> int:
+    """
+    Perform XOR multiplication of two integers, equivalent to polynomial
+    multiplication in GF(2).
+
+    >>> xor_multiply(3, 5)  # (011) ⊗ (101)
+    15
+    """
+    res = 0
+    while b:
+        if b & 1:
+            res ^= a
+        a <<= 1
+        b >>= 1
+    return res
+
+
+def divisors(n: int) -> set[int]:
+    """
+    Return all divisors of n (excluding 0).
+
+    >>> divisors(12)
+    {1, 2, 3, 4, 6, 12}
+    """
+    s = {1}
+    for i in range(2, int(n**0.5) + 1):
+        if n % i == 0:
+            s.add(i)
+            s.add(n // i)
+    s.add(n)
+    return s
+
+
+def mobius_table(n: int) -> list[int]:
+    """
+    Generate Möbius function values from 1 to n.
+
+    >>> mobius_table(10)[:6]
+    [0, 1, -1, -1, 0, -1]
+    """
+    mob = [1] * (n + 1)
+    is_prime = [True] * (n + 1)
+    mob[0] = 0
+
+    for p in range(2, n + 1):
+        if is_prime[p]:
+            mob[p] = -1
+            for j in range(2 * p, n + 1, p):
+                is_prime[j] = False
+                mob[j] *= -1
+            p2 = p * p
+            if p2 <= n:
+                for j in range(p2, n + 1, p2):
+                    mob[j] = 0
+    return mob
+
+
+def count_irreducibles(d: int) -> int:
+    """
+    Count the number of irreducible polynomials of degree d over GF(2)
+    using the Möbius function.
+
+    >>> count_irreducibles(3)
+    2
+    """
+    mob = mobius_table(d)
+    total = 0
+    for div in divisors(d) | {d}:
+        total += mob[div] * (1 << (d // div))
+    return total // d
+
+
+def find_xor_prime(rank: int) -> int:
+    """
+    Find the Nth XOR prime using a bitarray-based sieve.
+
+    >>> find_xor_prime(10)
+    41
+    """
+    total, degree = 0, 1
+    while True:
+        count = count_irreducibles(degree)
+        if total + count > rank:
+            break
+        total += count
+        degree += 1
+
+    limit = 1 << (degree + 1)
+
+    sieve = [True] * limit
+    sieve[0] = sieve[1] = False
+
+    for even in range(4, limit, 2):
+        sieve[even] = False
+
+    current = 1
+    for i in range(3, limit, 2):
+        if sieve[i]:
+            current += 1
+            if current == rank:
+                return i
+
+            j = i
+            while True:
+                prod = xor_multiply(i, j)
+                if prod >= limit:
+                    break
+                sieve[prod] = False
+                j += 2
+
+    raise ValueError(
+        "Failed to locate the requested XOR-prime within the computed limit"
+    )
+
+
+def solution(limit: int = 5000001) -> int:
+    """
+    Wrapper for Project Euler-style solution function.
+
+    >>> solution(10)
+    41
+    """
+    result = find_xor_prime(limit)
+    return result
+
+
+if __name__ == "__main__":
+    print(f"{solution(5000000) = }")