feat: add sieve of atkin algorithm (#6709)

* feat: add sieve of atkin algorithm

* fix: add full test coverage

* refactor: if condition braces

---------

Co-authored-by: a <alexanderklmn@gmail.com>

src/main/java/com/thealgorithms/maths/SieveOfAtkin.java

@@ -0,0 +1,143 @@
+package com.thealgorithms.maths;
+
+import java.util.ArrayList;
+import java.util.List;
+
+/**
+ * Implementation of the Sieve of Atkin, an optimized algorithm to generate
+ * all prime numbers up to a given limit.
+ *
+ * The Sieve of Atkin uses quadratic forms and modular arithmetic to identify
+ * prime candidates, then eliminates multiples of squares. It is more efficient
+ * than the Sieve of Eratosthenes for large limits.
+ */
+public final class SieveOfAtkin {
+
+    private SieveOfAtkin() {
+        // Utlity class; prevent instantiation
+    }
+
+    /**
+     * Generates a list of all prime numbers up to the specified limit
+     * using the Sieve of Atkin algorithm.
+     *
+     * @param limit the upper bound up to which primes are generated; must be zero or positive
+     * @return a list of prime numbers up to the limit; empty if the limit is less than 2
+     */
+    public static List<Integer> generatePrimes(int limit) {
+        if (limit < 1) {
+            return List.of();
+        }
+
+        boolean[] sieve = new boolean[limit + 1];
+        int sqrtLimit = (int) Math.sqrt(limit);
+
+        markQuadraticResidues(limit, sqrtLimit, sieve);
+        eliminateMultiplesOfSquares(limit, sqrtLimit, sieve);
+
+        List<Integer> primes = new ArrayList<>();
+        if (limit >= 2) {
+            primes.add(2);
+        }
+        if (limit >= 3) {
+            primes.add(3);
+        }
+
+        for (int i = 5; i <= limit; i++) {
+            if (sieve[i]) {
+                primes.add(i);
+            }
+        }
+
+        return primes;
+    }
+
+    /**
+     * Marks numbers in the sieve as prime candidates based on quadratic residues.
+     *
+     * This method iterates over all x and y up to sqrt(limit) and applies
+     * the three quadratic forms used in the Sieve of Atkin. Numbers satisfying
+     * the modulo conditions are toggled in the sieve array.
+     *
+     * @param limit the upper bound for primes
+     * @param sqrtLimit square root of the limit
+     * @param sieve boolean array representing potential primes
+     */
+    private static void markQuadraticResidues(int limit, int sqrtLimit, boolean[] sieve) {
+        for (int x = 1; x <= sqrtLimit; x++) {
+            for (int y = 1; y <= sqrtLimit; y++) {
+                applyQuadraticForm(4 * x * x + y * y, limit, sieve, 1, 5);
+                applyQuadraticForm(3 * x * x + y * y, limit, sieve, 7);
+                applyQuadraticForm(3 * x * x - y * y, limit, sieve, 11, x > y);
+            }
+        }
+    }
+
+    /**
+     * Toggles the sieve entry for a number if it satisfies one modulo condition.
+     *
+     * @param n the number to check
+     * @param limit upper bound of primes
+     * @param sieve boolean array representing potential primes
+     * @param modulo the modulo condition number to check
+     */
+    private static void applyQuadraticForm(int n, int limit, boolean[] sieve, int modulo) {
+        if (n <= limit && n % 12 == modulo) {
+            sieve[n] ^= true;
+        }
+    }
+
+    /**
+     * Toggles the sieve entry for a number if it satisfies either of two modulo conditions.
+     *
+     * @param n the number to check
+     * @param limit upper bound of primes
+     * @param sieve boolean array representing potential primes
+     * @param modulo1 first modulo condition number to check
+     * @param modulo2 second modulo condition number to check
+     */
+    private static void applyQuadraticForm(int n, int limit, boolean[] sieve, int modulo1, int modulo2) {
+        if (n <= limit && (n % 12 == modulo1 || n % 12 == modulo2)) {
+            sieve[n] ^= true;
+        }
+    }
+
+    /**
+     * Toggles the sieve entry for a number if it satisfies the modulo condition and an additional boolean condition.
+     *
+     * This version is used for the quadratic form 3*x*x - y*y, which requires x > y.
+     *
+     * @param n the number to check
+     * @param limit upper bound of primes
+     * @param sieve boolean array representing potential primes
+     * @param modulo the modulo condition number to check
+     * @param condition an additional boolean condition that must be true
+     */
+    private static void applyQuadraticForm(int n, int limit, boolean[] sieve, int modulo, boolean condition) {
+        if (condition && n <= limit && n % 12 == modulo) {
+            sieve[n] ^= true;
+        }
+    }
+
+    /**
+     * Eliminates numbers that are multiples of squares from the sieve.
+     *
+     * All numbers that are multiples of i*i (where i is marked as prime) are
+     * marked non-prime to finalize the sieve. This ensures only actual primes remain.
+     *
+     * @param limit the upper bound for primes
+     * @param sqrtLimit square root of the limit
+     * @param sieve boolean array representing potential primes
+     */
+    private static void eliminateMultiplesOfSquares(int limit, int sqrtLimit, boolean[] sieve) {
+        for (int i = 5; i <= sqrtLimit; i++) {
+            if (!sieve[i]) {
+                continue;
+            }
+            int square = i * i;
+            for (int j = square; j <= limit; j += square) {
+                sieve[j] = false;
+            }
+        }
+    }
+}