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0279-perfect-squares/0279-perfect-squares.py

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+class Solution:
+    def check_two(self, num : int) -> bool:
+        # CASE k = 2 : Fermat's Two Square Theorem
+        # divide the numbers as long as possible with
+        # the given bases `2, 5, 9`
+        bases = [2, 5, 9]
+        for base in bases:
+            while (num % base) == 0: num //= base
+
+        if (num % 3) == 0:
+            return False
+        elif (num in (0, 1, 13, 17)):
+            return True
+
+        i, j = 0, int(num ** 0.5)
+        while i <= j:
+            if (i * i) + (j * j) == num:
+                return True
+            elif (i * i) + (j * j) < num:
+                i += 1
+            elif (i * i) + (j * j) > num:
+                j -= 1
+            else:
+                continue
+
+        return False
+
+    def numSquares(self, n: int) -> int:
+        # https://leetcode.com/problems/perfect-squares/discuss/2837639
+        # https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem
+        i = 1
+
+        while (i * i) <= n:
+            # CASE k = 1 : Rudd's One Square Theorem
+            if (i * i) == n:
+                return 1
+
+            i += 1
+
+        # implement second case
+        if self.check_two(n):
+            return 2
+
+        # CASE k = 3 : Legendre's Three Square Theorem
+        while not (n % 4):
+            n //= 4
+
+        if (n % 8) != 7:
+            return 3
+
+        # CASE k = 4 : Legrange's Four Square Theorem
+        return 4