Implement UInt128 exponentiation and apply in Mersenne scan

PerfectNumbers.Core.Tests/UInt128Extensions/UInt128ExtensionsTests.cs

@@ -112,6 +112,40 @@ public class UInt128ExtensionsTests
         mod3.Should().Be(expected3);
     }
 
+    [Theory]
+    [Trait("Category", "Fast")]
+    [InlineData(2UL, 0UL)]
+    [InlineData(2UL, 1UL)]
+    [InlineData(2UL, 63UL)]
+    [InlineData(2UL, 127UL)]
+    [InlineData(3UL, 5UL)]
+    public void Pow_matches_BigInteger(ulong value, ulong exponent)
+    {
+        UInt128 v = value;
+        UInt128 expected = (UInt128)BigPow(value, exponent);
+
+        v.Pow(exponent).Should().Be(expected);
+    }
+
+    private static BigInteger BigPow(ulong value, ulong exponent)
+    {
+        BigInteger result = BigInteger.One;
+        BigInteger baseValue = value;
+
+        while (exponent != 0UL)
+        {
+            if ((exponent & 1UL) != 0UL)
+            {
+                result *= baseValue;
+            }
+
+            baseValue *= baseValue;
+            exponent >>= 1;
+        }
+
+        return result;
+    }
+
     [Theory]
     [Trait("Category", "Fast")]
     [InlineData(2UL, 10UL, 17UL)]

PerfectNumbers.Core/MersenneNumberTester.cs

@@ -214,7 +214,7 @@ public sealed class MersenneNumberTester(
 		bool prePrime = true;
         UInt128 twoP = (UInt128)exponent << 1; // 2 * p
         // UInt128 maxDivisor = new(ulong.MaxValue, ulong.MaxValue);
-        UInt128 maxK = (UInt128Numbers.Two ^ exponent) / 2 + 1;
+        UInt128 maxK = UInt128Numbers.Two.Pow(exponent) / 2 + 1;
         bool lastIsSeven = (exponent & 3UL) == 3UL;
 
 		// This is how maxK was for residue test before changes UInt128 maxK = maxKOverride ?? ResidueMaxK;

PerfectNumbers.Core/UInt128Extensions.cs

@@ -150,15 +150,35 @@ public static class UInt128Extensions
 			high >>= 64;
 		}
 
-		mod10 = result % 10UL;
+                mod10 = result % 10UL;
-	}
+        }
 
-	[MethodImpl(MethodImplOptions.AggressiveInlining)]
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
-	public static UInt128 ModPow(this UInt128 value, UInt128 exponent, UInt128 modulus)
+        public static UInt128 Pow(this UInt128 value, ulong exponent)
-	{
+        {
-		UInt128 result, one = result = UInt128.One,
+                UInt128 result = UInt128.One;
-				zero = UInt128.Zero,
+                UInt128 baseValue = value;
-				baseValue = value % modulus;
+
+                while (exponent != 0UL)
+                {
+                        if ((exponent & 1UL) != 0UL)
+                        {
+                                result *= baseValue;
+                        }
+
+                        baseValue *= baseValue;
+                        exponent >>= 1;
+                }
+
+                return result;
+        }
+
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        public static UInt128 ModPow(this UInt128 value, UInt128 exponent, UInt128 modulus)
+        {
+                UInt128 result, one = result = UInt128.One,
+                                zero = UInt128.Zero,
+                                baseValue = value % modulus;
 
 		while (exponent != zero)
 		{