Algorithms: Number Theory

Algorithms: Number Theory

Explore algorithms: number theory code snippets and tutorials

Algorithms: Number Theory

Segmented Sieve in Java

A segmented sieve marks composites only inside a requested interval, reducing memory compared with sieving all the way to its upper endpoint..

java 
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// Segmented Sieve in Java
//
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 6.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("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) {
        alg_require(((low >= 0) && (low <= high) && (high <= 1000000)));
        int limit = 1;
        while ((limit <= (high / limit))) {
            limit = Math.addExact(limit, 1);
        }
        int[] base = new int[alg_size(Math.addExact(limit, 1))];
        int[] marked = new int[alg_size(Math.addExact(Math.subtractExact(high, low), 1))];
        int p = 2;
        int multiple = 0;
        int count = 0;
        int i = 0;
        while ((p < limit)) {
            if ((base[p] == 0)) {
                multiple = Math.multiplyExact(p, p);
                while ((multiple < limit)) {
                    base[multiple] = 1;
                    multiple = Math.addExact(multiple, p);
                }
                multiple = Math.max(Math.multiplyExact(p, p), Math.multiplyExact((Math.subtractExact(Math.addExact(low, p), 1) / p), p));
                while ((multiple <= high)) {
                    marked[Math.subtractExact(multiple, low)] = 1;
                    multiple = Math.addExact(multiple, p);
                }
            }
            p = Math.addExact(p, 1);
        }
        i = Math.max(2, low);
        while ((i <= high)) {
            if ((marked[Math.subtractExact(i, low)] == 0)) {
                count = Math.addExact(count, 1);
            }
            i = Math.addExact(i, 1);
        }
        return count;
    }

    public static void main(String[] args) {
        int low = 10;
        int high = 30;
        int result = segmented_sieve(low, high);
        System.out.println(result);
    }
}
Nov 15, 2026 Java
Algorithms: Number Theory

Segmented Sieve in JavaScript

A segmented sieve marks composites only inside a requested interval, reducing memory compared with sieving all the way to its upper endpoint..

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"use strict";
// Segmented Sieve in JavaScript
//
// 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
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 6.
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function segmented_sieve(low, high) {
    alg_require(Number.isInteger(low) && low >= -2147483648 && low <= 2147483647);
    alg_require(Number.isInteger(high) && high >= -2147483648 && high <= 2147483647);
    alg_require(((low >= 0) && (low <= high) && (high <= 1000000)));
    let limit = 1;
    while ((limit <= Math.floor(high / limit))) {
        limit = alg_add(limit, 1);
    }
    let base = new Int32Array(alg_size(alg_add(limit, 1)));
    let marked = new Int32Array(alg_size(alg_add(alg_sub(high, low), 1)));
    let p = 2;
    let multiple = 0;
    let count = 0;
    let i = 0;
    while ((p < limit)) {
        if ((base[p] == 0)) {
            multiple = alg_mul(p, p);
            while ((multiple < limit)) {
                base[multiple] = 1;
                multiple = alg_add(multiple, p);
            }
            multiple = Math.max(alg_mul(p, p), alg_mul(Math.floor(alg_sub(alg_add(low, p), 1) / p), p));
            while ((multiple <= high)) {
                marked[alg_sub(multiple, low)] = 1;
                multiple = alg_add(multiple, p);
            }
        }
        p = alg_add(p, 1);
    }
    i = Math.max(2, low);
    while ((i <= high)) {
        if ((marked[alg_sub(i, low)] == 0)) {
            count = alg_add(count, 1);
        }
        i = alg_add(i, 1);
    }
    return count;
}

let low = 10;
let high = 30;
let result = segmented_sieve(low, high);
console.log(result);
Nov 15, 2026 javascript
Algorithms: Number Theory

Segmented Sieve in Python

A segmented sieve marks composites only inside a requested interval, reducing memory compared with sieving all the way to its upper endpoint..

python 
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from bisect import bisect_left

# Segmented Sieve in Python
#
# 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
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 6.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def segmented_sieve(low: int, high: int) -> int:
    alg_require(type(low) is int and -2147483648 <= low <= 2147483647)
    alg_require(type(high) is int and -2147483648 <= high <= 2147483647)
    alg_require(low >= 0 and low <= high and (high <= 1000000))
    limit: int = 1
    while limit <= high // limit:
        limit += 1
    base: list[int] = [0] * alg_size(limit + 1)
    marked: list[int] = [0] * alg_size(high - low + 1)
    p: int = 2
    multiple: int = 0
    count: int = 0
    i: int = 0
    while p < limit:
        if base[p] == 0:
            multiple = p * p
            while multiple < limit:
                base[multiple] = 1
                multiple += p
            multiple = max(p * p, (low + p - 1) // p * p)
            while multiple <= high:
                marked[multiple - low] = 1
                multiple += p
        p += 1
    i = max(2, low)
    while i <= high:
        if marked[i - low] == 0:
            count += 1
        i += 1
    return count

low = 10
high = 30
result = segmented_sieve(low, high)
print(result)
Nov 15, 2026 python
Algorithms: Number Theory

Segmented Sieve in C

A segmented sieve marks composites only inside a requested interval, reducing memory compared with sieving all the way to its upper endpoint..

c 
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#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// 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
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 6.
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int segmented_sieve(int low, int high) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *base = NULL;
    int *marked = NULL;
    if (!(((low >= 0) && (low <= high) && (high <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int limit = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (limit <= (high / limit));
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        limit = alg_add(limit, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(limit, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        base = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*base));
        if (!base) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(alg_sub(high, low, &alg_error), 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        marked = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*marked));
        if (!marked) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int p = 2;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int multiple = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int count = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (p < limit);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        int alg_condition_1 = (base[p] == 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_1) {
            multiple = alg_mul(p, p, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (multiple < limit);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                base[multiple] = 1;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                multiple = alg_add(multiple, p, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
            multiple = alg_max(alg_mul(p, p, &alg_error), alg_mul((alg_sub(alg_add(low, p, &alg_error), 1, &alg_error) / p), p, &alg_error));
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (multiple <= high);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                marked[alg_sub(multiple, low, &alg_error)] = 1;
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                multiple = alg_add(multiple, p, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
        }
        p = alg_add(p, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    i = alg_max(2, low);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (i <= high);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        int alg_condition_2 = (marked[alg_sub(i, low, &alg_error)] == 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_condition_2) {
            count = alg_add(count, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        i = alg_add(i, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = count;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(base);
    free(marked);
    return alg_result;
}

int main(void) {
    int low = 10;
    int high = 30;
    int result = segmented_sieve(low, high);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}
Nov 15, 2026 C
Algorithms: Number Theory

Segmented Sieve in C++

A segmented sieve marks composites only inside a requested interval, reducing memory compared with sieving all the way to its upper endpoint..

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#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// 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++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 6.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int segmented_sieve(int low, int high) {
    alg_require(((low >= 0) && (low <= high) && (high <= 1000000)));
    int limit = 1;
    while ((limit <= (high / limit))) {
        limit = alg_add(limit, 1);
    }
    std::vector<int> base(alg_size(alg_add(limit, 1)), 0);
    std::vector<int> marked(alg_size(alg_add(alg_sub(high, low), 1)), 0);
    int p = 2;
    int multiple = 0;
    int count = 0;
    int i = 0;
    while ((p < limit)) {
        if ((base[p] == 0)) {
            multiple = alg_mul(p, p);
            while ((multiple < limit)) {
                base[multiple] = 1;
                multiple = alg_add(multiple, p);
            }
            multiple = alg_max(alg_mul(p, p), alg_mul((alg_sub(alg_add(low, p), 1) / p), p));
            while ((multiple <= high)) {
                marked[alg_sub(multiple, low)] = 1;
                multiple = alg_add(multiple, p);
            }
        }
        p = alg_add(p, 1);
    }
    i = alg_max(2, low);
    while ((i <= high)) {
        if ((marked[alg_sub(i, low)] == 0)) {
            count = alg_add(count, 1);
        }
        i = alg_add(i, 1);
    }
    return count;
}

int main(void) {
    int low = 10;
    int high = 30;
    int result = segmented_sieve(low, high);
    printf("%d\n", result);
    return 0;
}
Nov 15, 2026 C++