Added Sum of Squares Lagrange's Four Square algorithm implementation (#6583)

* Add Sum of Squares algorithm implementation

* Format code and add Wikipedia URL for Lagrange's theorem

* Fixed clang-format issues

---------

Co-authored-by: Oleksandr Klymenko <[email protected]>

Author: BEASTSHRIRAM

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

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+package com.thealgorithms.maths;
+
+/**
+ * Implementation of Lagrange's Four Square Theorem
+ * Find minimum number of perfect squares that sum to given number
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's Four Square Theorem</a>
+ * @author BEASTSHRIRAM
+ */
+public final class SumOfSquares {
+
+    private SumOfSquares() {
+        // Utility class
+    }
+
+    /**
+     * Find minimum number of perfect squares that sum to n
+     *
+     * @param n the target number
+     * @return minimum number of squares needed
+     */
+    public static int minSquares(int n) {
+        if (isPerfectSquare(n)) {
+            return 1;
+        }
+
+        for (int i = 1; i * i <= n; i++) {
+            int remaining = n - i * i;
+            if (isPerfectSquare(remaining)) {
+                return 2;
+            }
+        }
+
+        // Legendre's three-square theorem
+        int temp = n;
+        while (temp % 4 == 0) {
+            temp /= 4;
+        }
+        if (temp % 8 == 7) {
+            return 4;
+        }
+
+        return 3;
+    }
+
+    private static boolean isPerfectSquare(int n) {
+        if (n < 0) {
+            return false;
+        }
+        int root = (int) Math.sqrt(n);
+        return root * root == n;
+    }
+}

src/test/java/com/thealgorithms/maths/SumOfSquaresTest.java

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+package com.thealgorithms.maths;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+
+import org.junit.jupiter.api.Test;
+
+/**
+ * Test class for SumOfSquares
+ *
+ * @author BEASTSHRIRAM
+ */
+class SumOfSquaresTest {
+
+    @Test
+    void testPerfectSquares() {
+        // Perfect squares should return 1
+        assertEquals(1, SumOfSquares.minSquares(1)); // 1^2
+        assertEquals(1, SumOfSquares.minSquares(4)); // 2^2
+        assertEquals(1, SumOfSquares.minSquares(9)); // 3^2
+        assertEquals(1, SumOfSquares.minSquares(16)); // 4^2
+        assertEquals(1, SumOfSquares.minSquares(25)); // 5^2
+    }
+
+    @Test
+    void testTwoSquares() {
+        // Numbers that can be expressed as sum of two squares
+        assertEquals(2, SumOfSquares.minSquares(2)); // 1^2 + 1^2
+        assertEquals(2, SumOfSquares.minSquares(5)); // 1^2 + 2^2
+        assertEquals(2, SumOfSquares.minSquares(8)); // 2^2 + 2^2
+        assertEquals(2, SumOfSquares.minSquares(10)); // 1^2 + 3^2
+        assertEquals(2, SumOfSquares.minSquares(13)); // 2^2 + 3^2
+    }
+
+    @Test
+    void testThreeSquares() {
+        // Numbers that require exactly three squares
+        assertEquals(3, SumOfSquares.minSquares(3)); // 1^2 + 1^2 + 1^2
+        assertEquals(3, SumOfSquares.minSquares(6)); // 1^2 + 1^2 + 2^2
+        assertEquals(3, SumOfSquares.minSquares(11)); // 1^2 + 1^2 + 3^2
+        assertEquals(3, SumOfSquares.minSquares(12)); // 2^2 + 2^2 + 2^2
+        assertEquals(3, SumOfSquares.minSquares(14)); // 1^2 + 2^2 + 3^2
+    }
+
+    @Test
+    void testFourSquares() {
+        // Numbers that require exactly four squares (form 4^a * (8b + 7))
+        assertEquals(4, SumOfSquares.minSquares(7)); // 1^2 + 1^2 + 1^2 + 2^2
+        assertEquals(4, SumOfSquares.minSquares(15)); // 1^2 + 1^2 + 2^2 + 3^2
+        assertEquals(4, SumOfSquares.minSquares(23)); // 1^2 + 1^2 + 3^2 + 3^2
+        assertEquals(4, SumOfSquares.minSquares(28)); // 4 * 7, so needs 4 squares
+        assertEquals(4, SumOfSquares.minSquares(31)); // 1^2 + 2^2 + 3^2 + 3^2
+    }
+
+    @Test
+    void testLargerNumbers() {
+        // Test some larger numbers
+        assertEquals(1, SumOfSquares.minSquares(100)); // 10^2
+        assertEquals(2, SumOfSquares.minSquares(65)); // 1^2 + 8^2
+        assertEquals(3, SumOfSquares.minSquares(19)); // 1^2 + 3^2 + 3^2
+        assertEquals(4, SumOfSquares.minSquares(60)); // 4 * 15, and 15 = 8*1 + 7
+    }
+
+    @Test
+    void testEdgeCases() {
+        // Test edge case
+        assertEquals(1, SumOfSquares.minSquares(0)); // 0^2
+    }
+}