Refactor IsPrime: rename to PrimalityCheck, add docs and 6k±1 explanation

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Author: williamfiset

README.md

@@ -248,7 +248,7 @@ $ java -cp classes com.williamfiset.algorithms.search.BinarySearch
 - [Greatest Common Divisor (GCD)](src/main/java/com/williamfiset/algorithms/math/Gcd.java) **- ~O(log(a + b))**
 - [Fast Fourier transform (quick polynomial multiplication)](src/main/java/com/williamfiset/algorithms/math/FastFourierTransform.java) **- O(nlog(n))**
 - [Fast Fourier transform (quick polynomial multiplication, complex numbers)](src/main/java/com/williamfiset/algorithms/math/FastFourierTransformComplexNumbers.java) **- O(nlog(n))**
-- [Primality check](src/main/java/com/williamfiset/algorithms/math/IsPrime.java) **- O(√n)**
+- [Primality check](src/main/java/com/williamfiset/algorithms/math/PrimalityCheck.java) **- O(√n)**
 - [Primality check (Rabin-Miller)](src/main/java/com/williamfiset/algorithms/math/RabinMillerPrimalityTest.py) **- O(k)**
 - [Least Common Multiple (LCM)](src/main/java/com/williamfiset/algorithms/math/Lcm.java) **- ~O(log(a + b))**
 - [Modular inverse](src/main/java/com/williamfiset/algorithms/math/ModularInverse.java) **- ~O(log(a + b))**

src/main/java/com/williamfiset/algorithms/math/BUILD

@@ -42,10 +42,10 @@ java_binary(
     runtime_deps = [":math"],
 )
 
-# bazel run //src/main/java/com/williamfiset/algorithms/math:IsPrime
+# bazel run //src/main/java/com/williamfiset/algorithms/math:PrimalityCheck
 java_binary(
-    name = "IsPrime",
-    main_class = "com.williamfiset.algorithms.math.IsPrime",
+    name = "PrimalityCheck",
+    main_class = "com.williamfiset.algorithms.math.PrimalityCheck",
     runtime_deps = [":math"],
 )
 

src/main/java/com/williamfiset/algorithms/math/IsPrime.java

@@ -0,0 +1,50 @@
+/**
+ * Tests whether a number is prime using trial division. Skips multiples of 2 and 3 by iterating
+ * over numbers of the form 6k ± 1.
+ *
+ * <p>Why 6k ± 1? Every integer falls into one of six residue classes mod 6:
+ *
+ * <pre>
+ *   6k + 0  →  divisible by 6 (= 2·3)     → not prime
+ *   6k + 1  →  not divisible by 2 or 3     → candidate ✓
+ *   6k + 2  →  divisible by 2              → not prime
+ *   6k + 3  →  divisible by 3              → not prime
+ *   6k + 4  →  divisible by 2              → not prime
+ *   6k + 5  →  not divisible by 2 or 3     → candidate ✓  (same as 6k - 1)
+ * </pre>
+ *
+ * After checking 2 and 3 directly, all remaining primes must be of the form 6k ± 1, so we only
+ * need to test divisibility by those candidates.
+ *
+ * <p>Time: O(√n)
+ *
+ * @author Micah Stairs, William Fiset
+ */
+package com.williamfiset.algorithms.math;
+
+public class PrimalityCheck {
+
+  /** Returns true if n is a prime number. */
+  public static boolean isPrime(long n) {
+    if (n < 2)
+      return false;
+    if (n == 2 || n == 3)
+      return true;
+    if (n % 2 == 0 || n % 3 == 0)
+      return false;
+    long limit = (long) Math.sqrt(n);
+    for (long i = 5; i <= limit; i += 6)
+      if (n % i == 0 || n % (i + 2) == 0)
+        return false;
+    return true;
+  }
+
+  public static void main(String[] args) {
+    System.out.println(isPrime(5));    // true
+    System.out.println(isPrime(31));   // true
+    System.out.println(isPrime(1433)); // true
+    System.out.println(isPrime(8763857775536878331L)); // true
+    System.out.println(isPrime(4));    // false
+    System.out.println(isPrime(15));   // false
+  }
+}