Segmented Sieve in C#

using System;
// Segmented Sieve in C#
//
// PURPOSE
// A segmented sieve marks composites only inside a requested interval, reducing memory
// compared with sieving all the way to its upper endpoint.
//
// METHOD
// Sieve base primes up to square root of high. For each prime, mark its multiples in
// [low,high], starting no earlier than p squared. Count unmarked values at least two.
//
// COMPLEXITY
// O(sqrt(high) log log high + (high-low+1) log log high) time, O(sqrt(high)+high-low+1)
// space.
//
// CONTRACT
// 0 <= low <= high <= 1000000. Returns number of primes in the inclusive interval.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 6.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int segmented_sieve(int low, int high) {
        checked {
            alg_require(((low >= 0) && (low <= high) && (high <= 1000000)));
            int limit = 1;
            while ((limit <= (high / limit))) {
                limit = (limit + 1);
            }
            int[] base_value = new int[alg_size((limit + 1))];
            int[] marked = new int[alg_size(((high - low) + 1))];
            int p = 2;
            int multiple = 0;
            int count = 0;
            int i = 0;
            while ((p < limit)) {
                if ((base_value[p] == 0)) {
                    multiple = (p * p);
                    while ((multiple < limit)) {
                        base_value[multiple] = 1;
                        multiple = (multiple + p);
                    }
                    multiple = Math.Max((p * p), ((((low + p) - 1) / p) * p));
                    while ((multiple <= high)) {
                        marked[(multiple - low)] = 1;
                        multiple = (multiple + p);
                    }
                }
                p = (p + 1);
            }
            i = Math.Max(2, low);
            while ((i <= high)) {
                if ((marked[(i - low)] == 0)) {
                    count = (count + 1);
                }
                i = (i + 1);
            }
            return count;
        }
    }

    static void Main() {
        int low = 10;
        int high = 30;
        int result = segmented_sieve(low, high);
        Console.WriteLine(result);
    }
}