Algorithms: Number Theory

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);
    }
}

<?php
declare(strict_types=1);
// Segmented Sieve in PHP
//
// 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
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 6.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function segmented_sieve(int $low, int $high): int {
    alg_require((($low >= 0) && ($low <= $high) && ($high <= 1000000)));
    $limit = 1;
    while (($limit <= intdiv($high, $limit))) {
        $limit = alg_add($limit, 1);
    }
    $base = array_fill(0, alg_size(alg_add($limit, 1)), 0);
    $marked = array_fill(0, alg_size(alg_add(alg_sub($high, $low), 1)), 0);
    $p = 2;
    $multiple = 0;
    $count = 0;
    $i = 0;
    while (($p < $limit)) {
        if (($base[$p] == 0)) {
            $multiple = alg_mul($p, $p);
            while (($multiple < $limit)) {
                $base[$multiple] = 1;
                $multiple = alg_add($multiple, $p);
            }
            $multiple = max(alg_mul($p, $p), alg_mul(intdiv(alg_sub(alg_add($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 = 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;
}

$low = 10;
$high = 30;
$result = segmented_sieve($low, $high);
echo $result, PHP_EOL;

// Euler Totient by Factorization in Java
//
// PURPOSE
// Euler totient counts positive integers up to n that are coprime with n.
//
// METHOD
// For each distinct prime divisor p, remove all factors p from a working value and replace
// result with result-result/p. Handle any remaining prime factor.
//
// COMPLEXITY
// O(sqrt(n)) trial-division time and O(1) space.
//
// CONTRACT
// n >= 1. phi(1)=1.
// 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 totient(int n) {
        alg_require((n >= 1));
        int result = n;
        int p = 2;
        while ((p <= (n / p))) {
            if (((n % p) == 0)) {
                while (((n % p) == 0)) {
                    n = (n / p);
                }
                result = Math.subtractExact(result, (result / p));
            }
            p = Math.addExact(p, 1);
        }
        if ((n > 1)) {
            result = Math.subtractExact(result, (result / n));
        }
        return result;
    }

    public static void main(String[] args) {
        int n = 9;
        int result = totient(n);
        System.out.println(result);
    }
}

"use strict";
// Euler Totient by Factorization in JavaScript
//
// PURPOSE
// Euler totient counts positive integers up to n that are coprime with n.
//
// METHOD
// For each distinct prime divisor p, remove all factors p from a working value and replace
// result with result-result/p. Handle any remaining prime factor.
//
// COMPLEXITY
// O(sqrt(n)) trial-division time and O(1) space.
//
// CONTRACT
// n >= 1. phi(1)=1.
// 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 totient(n) {
    alg_require(Number.isInteger(n) && n >= -2147483648 && n <= 2147483647);
    alg_require((n >= 1));
    let result = n;
    let p = 2;
    while ((p <= Math.floor(n / p))) {
        if (((n % p) == 0)) {
            while (((n % p) == 0)) {
                n = Math.floor(n / p);
            }
            result = alg_sub(result, Math.floor(result / p));
        }
        p = alg_add(p, 1);
    }
    if ((n > 1)) {
        result = alg_sub(result, Math.floor(result / n));
    }
    return result;
}

let n = 9;
let result = totient(n);
console.log(result);

from bisect import bisect_left

# Euler Totient by Factorization in Python
#
# PURPOSE
# Euler totient counts positive integers up to n that are coprime with n.
#
# METHOD
# For each distinct prime divisor p, remove all factors p from a working value and replace
# result with result-result/p. Handle any remaining prime factor.
#
# COMPLEXITY
# O(sqrt(n)) trial-division time and O(1) space.
#
# CONTRACT
# n >= 1. phi(1)=1.
# 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 totient(n: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(n >= 1)
    result: int = n
    p: int = 2
    while p <= n // p:
        if n % p == 0:
            while n % p == 0:
                n = n // p
            result -= result // p
        p += 1
    if n > 1:
        result -= result // n
    return result

n = 9
result = totient(n)
print(result)