Merge pull request #130 from mlemanczyk/codex/optimize-number-types-for-performance

Add native-width Pollard-Brent variants

EvenPerfectBitScanner/Program.cs

@@ -515,9 +515,7 @@ internal static class Program
         {
             _byDivisorTester = mersenneOnGpu
                     ? new MersenneNumberDivisorByDivisorGpuTester()
-                    : new MersenneNumberDivisorByDivisorCpuTester(
-                        useGpuPow2Mod: !GpuContextPool.ForceCpu,
-                        useGpuCycleComputation: orderOnGpu);
+                    : new MersenneNumberDivisorByDivisorCpuTester();
             _byDivisorTester.BatchSize = scanBatchSize;
         }
 

PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs

@@ -3,7 +3,6 @@ using System.Collections.Concurrent;
 using System.Collections.Generic;
 using System.Numerics;
 using PerfectNumbers.Core;
-using PerfectNumbers.Core.Gpu;
 
 namespace PerfectNumbers.Core.Cpu;
 
@@ -11,18 +10,13 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
 {
     private readonly object _sync = new();
     private readonly ConcurrentBag<DivisorScanSession> _sessionPool = new();
-    private readonly bool _useGpuPow2Mod;
-    private readonly bool _useGpuCycleComputation;
     private ulong _divisorLimit;
     private ulong _lastStatusDivisor;
     private bool _isConfigured;
     private int _batchSize = 1_024;
+    private const int PollardBrentMaxRestarts = 4;
+    private const int PollardBrentIterationBudget = 262_144;
 
-    public MersenneNumberDivisorByDivisorCpuTester(bool useGpuPow2Mod = false, bool useGpuCycleComputation = false)
-    {
-        _useGpuPow2Mod = useGpuPow2Mod;
-        _useGpuCycleComputation = useGpuCycleComputation;
-    }
 
     public int BatchSize
     {
@@ -182,6 +176,31 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
             return false;
         }
 
+        ulong pollardIterationCount = 0UL;
+        if (prime > 3UL)
+        {
+            var pollardBrent = new MersennePollardBrent(prime, _batchSize);
+            if (pollardBrent.IsSupported)
+            {
+                bool pollardFound = pollardBrent.TryFindFactor(
+                    allowedMax,
+                    PollardBrentMaxRestarts,
+                    PollardBrentIterationBudget,
+                    out ulong pollardFactor,
+                    out pollardIterationCount);
+
+                if (pollardFound)
+                {
+                    processedCount = pollardIterationCount;
+                    lastProcessed = pollardFactor;
+                    processedAll = true;
+                    return true;
+                }
+            }
+        }
+
+        processedCount = pollardIterationCount;
+
         UInt128 step = (UInt128)prime << 1;
         if (step == UInt128.Zero)
         {
@@ -198,8 +217,6 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
         }
 
         Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>? factorCache = null;
-        Dictionary<ulong, ulong>? cycleRemainderCache = null;
-        CycleHint cycleHint = default;
         DivisorCycleCache cycleCache = DivisorCycleCache.Shared;
 
         byte step10 = (byte)(step % 10UL);
@@ -233,14 +250,35 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
             if (admissible && (remainder8 == 1 || remainder8 == 7) && remainder3 != 0 && remainder5 != 0 && remainder7 != 0 && remainder11 != 0)
             {
                 MontgomeryDivisorData divisorData = MontgomeryDivisorDataCache.Get(candidate);
-                if (TryResolveDivisorCycle(candidate, prime, divisorData, cycleCache, ref factorCache, ref cycleHint, out ulong divisorCycle) && divisorCycle != 0UL)
+                ulong divisorCycle;
+
+                if (candidate <= PerfectNumberConstants.MaxQForDivisorCycles)
                 {
-                    if (CheckDivisor(prime, candidate, divisorCycle, divisorData, ref cycleRemainderCache, ref cycleHint) != 0)
+                    divisorCycle = cycleCache.GetCycleLength(candidate);
+                }
+                else
+                {
+                    factorCache ??= new Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>(8);
+                    if (!MersenneDivisorCycles.TryCalculateCycleLengthForExponent(candidate, prime, divisorData, factorCache, out divisorCycle) || divisorCycle == 0UL)
                     {
-                        processedAll = true;
-                        return true;
+                        divisorCycle = cycleCache.GetCycleLength(candidate);
                     }
                 }
+
+				if (divisorCycle != 0 && divisorCycle == prime)
+				{
+					processedAll = true;
+					return true;
+				}
+				else if (divisorCycle == 0)
+				{
+					Console.WriteLine($"Divisor cycle wasn't calculated for ${prime}");
+				}				
+				// if (divisorCycle != 0UL && CheckDivisor(prime, divisorCycle, divisorData) != 0)
+				// {
+				// 	processedAll = true;
+				// 	return true;
+				// }
             }
 
             divisor += step;
@@ -256,152 +294,12 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
         return false;
     }
 
-    private byte CheckDivisor(
-        ulong prime,
-        ulong divisor,
-        ulong divisorCycle,
-        in MontgomeryDivisorData divisorData,
-        ref Dictionary<ulong, ulong>? cycleRemainders,
-        ref CycleHint cycleHint)
+    private static byte CheckDivisor(ulong prime, ulong divisorCycle, in MontgomeryDivisorData divisorData)
     {
-        if (divisorCycle == 0UL)
-        {
-            throw new InvalidOperationException($"Cycle length resolution returned zero for divisor {divisor}.");
-        }
-
-        Dictionary<ulong, ulong>? remainderCache = cycleRemainders;
-        if (_useGpuPow2Mod && remainderCache is null)
-        {
-            remainderCache = new Dictionary<ulong, ulong>(8);
-            cycleRemainders = remainderCache;
-        }
-
-        ulong reducedExponent;
-        if (remainderCache is not null && remainderCache.TryGetValue(divisorCycle, out ulong cachedRemainder))
-        {
-            reducedExponent = cachedRemainder;
-        }
-        else if (cycleHint.HasCycle && cycleHint.LastCycleLength == divisorCycle)
-        {
-            reducedExponent = cycleHint.LastReducedExponent;
-        }
-        else
-        {
-            reducedExponent = prime % divisorCycle;
-            if (remainderCache is not null)
-            {
-                remainderCache[divisorCycle] = reducedExponent;
-            }
-        }
-
-        cycleHint.LastCycleLength = divisorCycle;
-        cycleHint.LastReducedExponent = reducedExponent;
-        cycleHint.HasCycle = true;
-
-        if (_useGpuPow2Mod && remainderCache is not null)
-        {
-            GpuPow2ModStatus status = PrimeOrderGpuHeuristics.TryPow2Mod(reducedExponent, divisor, out ulong remainder, divisorData);
-            if (status == GpuPow2ModStatus.Success)
-            {
-                return remainder == 1UL ? (byte)1 : (byte)0;
-            }
-
-            PrimeOrderGpuHeuristics.ReportPow2Failure(
-                "by-divisor residue scan",
-                divisor,
-                status);
-        }
-
-        ulong residue = reducedExponent.Pow2MontgomeryModFromCycleRemainder(divisorData);
+        ulong residue = prime.Pow2MontgomeryModWithCycle(divisorCycle, divisorData);
         return residue == 1UL ? (byte)1 : (byte)0;
     }
 
-    private bool TryResolveDivisorCycle(
-        ulong divisor,
-        ulong prime,
-        in MontgomeryDivisorData divisorData,
-        DivisorCycleCache cycleCache,
-        ref Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>? factorCache,
-        ref CycleHint cycleHint,
-        out ulong cycleLength)
-    {
-        if (divisor <= PerfectNumberConstants.MaxQForDivisorCycles)
-        {
-            cycleLength = cycleCache.GetCycleLength(divisor);
-            if (cycleLength == 0UL)
-            {
-                throw new InvalidOperationException($"Missing cached cycle length for divisor {divisor}.");
-            }
-
-            cycleHint.LastCycleLength = cycleLength;
-            cycleHint.HasCycle = true;
-            return true;
-        }
-
-        if (_useGpuCycleComputation && TryCalculateCycleLengthGpu(divisor, prime, divisorData, ref cycleHint, out cycleLength))
-        {
-            return cycleLength != 0UL;
-        }
-
-        factorCache ??= new Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>(8);
-        if (MersenneDivisorCycles.TryCalculateCycleLengthForExponent(
-                divisor,
-                prime,
-                divisorData,
-                factorCache,
-                out cycleLength,
-                allowGpuPow2: _useGpuPow2Mod) && cycleLength != 0UL)
-        {
-            cycleHint.LastCycleLength = cycleLength;
-            cycleHint.HasCycle = true;
-            return true;
-        }
-
-        cycleLength = cycleCache.GetCycleLength(divisor);
-        if (cycleLength == 0UL)
-        {
-            return false;
-        }
-
-        cycleHint.LastCycleLength = cycleLength;
-        cycleHint.HasCycle = true;
-        return true;
-    }
-
-    private bool TryCalculateCycleLengthGpu(
-        ulong divisor,
-        ulong prime,
-        in MontgomeryDivisorData divisorData,
-        ref CycleHint cycleHint,
-        out ulong cycleLength)
-    {
-        PrimeOrderCalculator.PrimeOrderSearchConfig config = PrimeOrderCalculator.PrimeOrderSearchConfig.HeuristicDefault;
-        ulong? previousOrder = cycleHint.HasCycle ? cycleHint.LastCycleLength : null;
-
-        try
-        {
-            if (!PrimeOrderGpuHeuristics.TryCalculateOrder(divisor, previousOrder, config, divisorData, out PrimeOrderCalculator.PrimeOrderResult result))
-            {
-                throw new InvalidOperationException("GPU requested fallback.");
-            }
-
-            cycleLength = result.Order;
-            if (cycleLength == 0UL)
-            {
-                return false;
-            }
-
-            cycleHint.LastCycleLength = cycleLength;
-            cycleHint.HasCycle = true;
-            return true;
-        }
-        catch (InvalidOperationException ex)
-        {
-            Console.Error.WriteLine($"GPU cycle length computation failed for divisor {divisor}: {ex.Message}");
-            throw;
-        }
-    }
-
     private static byte AddMod(byte value, byte delta, byte modulus)
     {
         int sum = value + delta;
@@ -448,13 +346,6 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
         return (1UL << (int)(maxPrime - 1UL)) - 1UL;
     }
 
-    private struct CycleHint
-    {
-        public ulong LastCycleLength;
-        public ulong LastReducedExponent;
-        public bool HasCycle;
-    }
-
     private static ulong ComputeAllowedMaxDivisor(ulong prime, ulong divisorLimit)
     {
         if (prime <= 1UL)

PerfectNumbers.Core/Cpu/MersennePollardBrent.cs

@@ -0,0 +1,926 @@
+using System;
+using System.Buffers;
+using System.Numerics;
+using System.Security.Cryptography;
+using PerfectNumbers.Core;
+using UInt128 = System.UInt128;
+
+namespace PerfectNumbers.Core.Cpu;
+
+internal sealed class MersennePollardBrent
+{
+    private enum ArithmeticMode
+    {
+        Unsupported,
+        UInt64,
+        UInt128,
+        BigInteger,
+    }
+
+    private readonly ArithmeticMode _mode;
+    private readonly int _bitLength;
+    private readonly int _byteLength;
+    private readonly int _bitRemainder;
+    private readonly byte _maskByte;
+    private readonly BigInteger _modulus;
+    private readonly BigInteger _mask;
+    private readonly ulong _modulus64;
+    private readonly ulong _mask64;
+    private readonly UInt128 _modulus128;
+    private readonly UInt128 _mask128;
+    private readonly int _batchSize;
+    private readonly bool _isSupported;
+
+    public MersennePollardBrent(ulong exponent, int batchSize)
+    {
+        if (exponent < 3UL || exponent > int.MaxValue)
+        {
+            _mode = ArithmeticMode.Unsupported;
+            _isSupported = false;
+            _bitLength = 0;
+            _byteLength = 0;
+            _bitRemainder = 0;
+            _maskByte = 0;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            _modulus64 = 0UL;
+            _mask64 = 0UL;
+            _modulus128 = UInt128.Zero;
+            _mask128 = UInt128.Zero;
+            _batchSize = 0;
+            return;
+        }
+
+        _isSupported = true;
+        _bitLength = (int)exponent;
+        _byteLength = (_bitLength + 7) / 8;
+        _bitRemainder = _bitLength & 7;
+        _maskByte = _bitRemainder == 0 ? (byte)0xFF : (byte)((1 << _bitRemainder) - 1);
+        _batchSize = Math.Clamp(batchSize, 8, 1024);
+
+        if (_bitLength <= 64)
+        {
+            _mode = ArithmeticMode.UInt64;
+            if (_bitLength == 64)
+            {
+                _modulus64 = ulong.MaxValue;
+                _mask64 = ulong.MaxValue;
+            }
+            else
+            {
+                _modulus64 = (1UL << _bitLength) - 1UL;
+                _mask64 = _modulus64;
+            }
+
+            _modulus128 = UInt128.Zero;
+            _mask128 = UInt128.Zero;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            return;
+        }
+
+        if (_bitLength <= 128)
+        {
+            _mode = ArithmeticMode.UInt128;
+            if (_bitLength == 128)
+            {
+                _modulus128 = UInt128.MaxValue;
+                _mask128 = UInt128.MaxValue;
+            }
+            else
+            {
+                _modulus128 = (UInt128.One << _bitLength) - UInt128.One;
+                _mask128 = _modulus128;
+            }
+
+            _modulus64 = 0UL;
+            _mask64 = 0UL;
+            _modulus = BigInteger.Zero;
+            _mask = BigInteger.Zero;
+            return;
+        }
+
+        _mode = ArithmeticMode.BigInteger;
+        _modulus = (BigInteger.One << _bitLength) - BigInteger.One;
+        _mask = _modulus;
+        _modulus64 = 0UL;
+        _mask64 = 0UL;
+        _modulus128 = UInt128.Zero;
+        _mask128 = UInt128.Zero;
+    }
+
+    public bool IsSupported => _isSupported;
+
+    public bool TryFindFactor(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        if (!_isSupported)
+        {
+            return false;
+        }
+
+        if (allowedMax <= 1UL)
+        {
+            return false;
+        }
+
+        return _mode switch
+        {
+            ArithmeticMode.UInt64 => TryFindFactorUInt64(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            ArithmeticMode.UInt128 => TryFindFactorUInt128(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            ArithmeticMode.BigInteger => TryFindFactorBigInteger(allowedMax, maxRestarts, maxIterationsPerRestart, out factor, out iterationCount),
+            _ => false,
+        };
+    }
+
+    private bool TryFindFactorUInt64(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            ulong x = NextRandomSeed64();
+            ulong y = x;
+            ulong c = NextRandomConstant64();
+            ulong q = 1UL;
+            ulong g = 1UL;
+            ulong ys = 0UL;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g == 1UL && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = Iterate64(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g == 1UL && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = Iterate64(y, c, ref localIterations);
+                        ulong diff = AbsSub64(y, x);
+                        if (diff == 0UL)
+                        {
+                            continue;
+                        }
+
+                        q = MulMod64(q, diff);
+                    }
+
+                    g = BinaryGcd64(q, _modulus64);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus64)
+            {
+                do
+                {
+                    ys = Iterate64(ys, c, ref localIterations);
+                    ulong diff = AbsSub64(ys, x);
+                    if (diff == 0UL)
+                    {
+                        continue;
+                    }
+
+                    g = BinaryGcd64(diff, _modulus64);
+                }
+                while (g == 1UL);
+            }
+
+            iterationCount += localIterations;
+
+            if (g == 1UL || g == _modulus64)
+            {
+                continue;
+            }
+
+            if (g > 1UL && g <= allowedMax)
+            {
+                factor = g;
+                return true;
+            }
+        }
+
+        return false;
+    }
+
+    private bool TryFindFactorUInt128(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+        UInt128 allowedMaxWide = (UInt128)allowedMax;
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            UInt128 x = NextRandomSeed128();
+            UInt128 y = x;
+            UInt128 c = NextRandomConstant128();
+            UInt128 q = UInt128.One;
+            UInt128 g = UInt128.One;
+            UInt128 ys = UInt128.Zero;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g == UInt128.One && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = Iterate128(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g == UInt128.One && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = Iterate128(y, c, ref localIterations);
+                        UInt128 diff = AbsSub128(y, x);
+                        if (diff == UInt128.Zero)
+                        {
+                            continue;
+                        }
+
+                        q = MulMod128(q, diff);
+                    }
+
+                    g = q.BinaryGcd(_modulus128);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus128)
+            {
+                do
+                {
+                    ys = Iterate128(ys, c, ref localIterations);
+                    UInt128 diff = AbsSub128(ys, x);
+                    if (diff == UInt128.Zero)
+                    {
+                        continue;
+                    }
+
+                    g = diff.BinaryGcd(_modulus128);
+                }
+                while (g == UInt128.One);
+            }
+
+            iterationCount += localIterations;
+
+            if (g == UInt128.One || g == _modulus128)
+            {
+                continue;
+            }
+
+            if (g > UInt128.One && g <= allowedMaxWide)
+            {
+                factor = (ulong)g;
+                return true;
+            }
+        }
+
+        return false;
+    }
+
+    private bool TryFindFactorBigInteger(
+        ulong allowedMax,
+        int maxRestarts,
+        int maxIterationsPerRestart,
+        out ulong factor,
+        out ulong iterationCount)
+    {
+        factor = 0UL;
+        iterationCount = 0UL;
+
+        int restarts = Math.Max(1, maxRestarts);
+        int iterationBudget = Math.Max(_batchSize, maxIterationsPerRestart);
+
+        for (int restart = 0; restart < restarts; restart++)
+        {
+            BigInteger x = NextRandomSeedBigInteger();
+            BigInteger y = x;
+            BigInteger c = NextRandomConstantBigInteger();
+            BigInteger q = BigInteger.One;
+            BigInteger g = BigInteger.One;
+            BigInteger ys = BigInteger.Zero;
+            int r = 1;
+            ulong localIterations = 0UL;
+
+            while (g.IsOne && localIterations < (ulong)iterationBudget)
+            {
+                x = y;
+                for (int i = 0; i < r && localIterations < (ulong)iterationBudget; i++)
+                {
+                    y = IterateBigInteger(y, c, ref localIterations);
+                }
+
+                if (localIterations >= (ulong)iterationBudget)
+                {
+                    break;
+                }
+
+                int k = 0;
+                while (k < r && g.IsOne && localIterations < (ulong)iterationBudget)
+                {
+                    ys = y;
+                    int t = Math.Min(_batchSize, r - k);
+                    for (int i = 0; i < t; i++)
+                    {
+                        y = IterateBigInteger(y, c, ref localIterations);
+                        BigInteger diff = AbsSubBigInteger(y, x);
+                        if (diff.IsZero)
+                        {
+                            continue;
+                        }
+
+                        q = MulModBigInteger(q, diff);
+                    }
+
+                    g = BigInteger.GreatestCommonDivisor(q, _modulus);
+                    k += t;
+                }
+
+                r <<= 1;
+            }
+
+            if (g == _modulus)
+            {
+                do
+                {
+                    ys = IterateBigInteger(ys, c, ref localIterations);
+                    BigInteger diff = AbsSubBigInteger(ys, x);
+                    if (diff.IsZero)
+                    {
+                        continue;
+                    }
+
+                    g = BigInteger.GreatestCommonDivisor(diff, _modulus);
+                }
+                while (g.IsOne);
+            }
+
+            iterationCount += localIterations;
+
+            if (g.IsOne || g == _modulus)
+            {
+                continue;
+            }
+
+            if (g <= ulong.MaxValue)
+            {
+                ulong candidate = (ulong)g;
+                if (candidate > 1UL && candidate <= allowedMax)
+                {
+                    factor = candidate;
+                    return true;
+                }
+            }
+        }
+
+        return false;
+    }
+
+    private ulong Iterate64(ulong value, ulong c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareMod64(value);
+        value = AddMod64(value, c);
+        return value;
+    }
+
+    private UInt128 Iterate128(UInt128 value, UInt128 c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareMod128(value);
+        value = AddMod128(value, c);
+        return value;
+    }
+
+    private BigInteger IterateBigInteger(BigInteger value, BigInteger c, ref ulong iterationCounter)
+    {
+        iterationCounter++;
+        value = SquareModBigInteger(value);
+        value = AddModBigInteger(value, c);
+        return value;
+    }
+
+    private ulong AddMod64(ulong a, ulong b)
+    {
+        UInt128 sum = (UInt128)a + b;
+        return Reduce64(sum);
+    }
+
+    private UInt128 AddMod128(UInt128 a, UInt128 b)
+    {
+        UInt128 sum = a + b;
+        bool carryOut = sum < a;
+        if (!carryOut && sum < _modulus128)
+        {
+            return sum;
+        }
+
+        ulong p3 = 0UL;
+        ulong p2 = carryOut ? 1UL : 0UL;
+        ulong p1 = (ulong)(sum >> 64);
+        ulong p0 = (ulong)sum;
+        return ReduceProductBitwise(p3, p2, p1, p0, _modulus128);
+    }
+
+    private BigInteger AddModBigInteger(BigInteger a, BigInteger b)
+    {
+        return ReduceBigInteger(a + b);
+    }
+
+    private ulong SquareMod64(ulong value)
+    {
+        UInt128 square = (UInt128)value * value;
+        return Reduce64(square);
+    }
+
+    private UInt128 SquareMod128(UInt128 value)
+    {
+        return MulMod128(value, value);
+    }
+
+    private BigInteger SquareModBigInteger(BigInteger value)
+    {
+        return ReduceBigInteger(value * value);
+    }
+
+    private ulong MulMod64(ulong a, ulong b)
+    {
+        UInt128 product = (UInt128)a * b;
+        return Reduce64(product);
+    }
+
+    private UInt128 MulMod128(UInt128 a, UInt128 b)
+    {
+        MultiplyFull(a, b, out ulong p3, out ulong p2, out ulong p1, out ulong p0);
+        return ReduceProductBitwise(p3, p2, p1, p0, _modulus128);
+    }
+
+    private BigInteger MulModBigInteger(BigInteger a, BigInteger b)
+    {
+        if (b == _modulus)
+        {
+            return a;
+        }
+
+        if (a == _modulus)
+        {
+            a = BigInteger.Zero;
+        }
+
+        return ReduceBigInteger(a * b);
+    }
+
+    private ulong Reduce64(UInt128 value)
+    {
+        if (_bitLength == 64)
+        {
+            UInt128 combined = (UInt128)(ulong)value + (value >> 64);
+            UInt128 modulus = _modulus64;
+            if (combined >= modulus)
+            {
+                combined -= modulus;
+                if (combined >= modulus)
+                {
+                    combined -= modulus;
+                }
+            }
+
+            return (ulong)combined;
+        }
+
+        UInt128 mask = _mask64;
+        UInt128 modulusValue = _modulus64;
+        UInt128 working = value;
+        while (true)
+        {
+            UInt128 high = working >> _bitLength;
+            if (high == UInt128.Zero)
+            {
+                break;
+            }
+
+            working = (working & mask) + high;
+        }
+
+        if (working >= modulusValue)
+        {
+            working -= modulusValue;
+            if (working >= modulusValue)
+            {
+                working %= modulusValue;
+                if (working >= modulusValue)
+                {
+                    working -= modulusValue;
+                }
+            }
+        }
+
+        return (ulong)working;
+    }
+
+    private BigInteger ReduceBigInteger(BigInteger value)
+    {
+        if (value.Sign < 0)
+        {
+            value %= _modulus;
+            if (value.Sign < 0)
+            {
+                value += _modulus;
+            }
+
+            return value;
+        }
+
+        while (true)
+        {
+            BigInteger high = value >> _bitLength;
+            if (high.IsZero)
+            {
+                break;
+            }
+
+            value &= _mask;
+            value += high;
+        }
+
+        if (value >= _modulus)
+        {
+            value -= _modulus;
+            if (value >= _modulus)
+            {
+                value %= _modulus;
+                if (value.Sign < 0)
+                {
+                    value += _modulus;
+                }
+            }
+        }
+
+        return value;
+    }
+
+    private ulong NextRandomSeed64()
+    {
+        ulong value;
+        do
+        {
+            value = NextRandomBelowModulus64();
+        }
+        while (value <= 1UL);
+
+        return value;
+    }
+
+    private UInt128 NextRandomSeed128()
+    {
+        UInt128 value;
+        do
+        {
+            value = NextRandomBelowModulus128();
+        }
+        while (value <= UInt128.One);
+
+        return value;
+    }
+
+    private BigInteger NextRandomSeedBigInteger()
+    {
+        BigInteger value;
+        do
+        {
+            value = NextRandomBelowModulusBigInteger();
+        }
+        while (value <= BigInteger.One);
+
+        return value;
+    }
+
+    private ulong NextRandomConstant64()
+    {
+        ulong value;
+        do
+        {
+            value = NextRandomBelowModulus64();
+        }
+        while (value == 0UL);
+
+        return value;
+    }
+
+    private UInt128 NextRandomConstant128()
+    {
+        UInt128 value;
+        do
+        {
+            value = NextRandomBelowModulus128();
+        }
+        while (value == UInt128.Zero);
+
+        return value;
+    }
+
+    private BigInteger NextRandomConstantBigInteger()
+    {
+        BigInteger value;
+        do
+        {
+            value = NextRandomBelowModulusBigInteger();
+        }
+        while (value.IsZero);
+
+        return value;
+    }
+
+    private ulong NextRandomBelowModulus64()
+    {
+        if (_byteLength == 0)
+        {
+            return 0UL;
+        }
+
+        Span<byte> buffer = stackalloc byte[_byteLength];
+        ulong candidate;
+        do
+        {
+            RandomNumberGenerator.Fill(buffer);
+            if (_bitRemainder != 0)
+            {
+                buffer[_byteLength - 1] &= _maskByte;
+            }
+
+            candidate = (ulong)ReadUInt128LittleEndian(buffer);
+        }
+        while (candidate >= _modulus64);
+
+        return candidate;
+    }
+
+    private UInt128 NextRandomBelowModulus128()
+    {
+        if (_byteLength == 0)
+        {
+            return UInt128.Zero;
+        }
+
+        Span<byte> buffer = stackalloc byte[_byteLength];
+        UInt128 candidate;
+        do
+        {
+            RandomNumberGenerator.Fill(buffer);
+            if (_bitRemainder != 0)
+            {
+                buffer[_byteLength - 1] &= _maskByte;
+            }
+
+            candidate = ReadUInt128LittleEndian(buffer);
+        }
+        while (candidate >= _modulus128);
+
+        return candidate;
+    }
+
+    private BigInteger NextRandomBelowModulusBigInteger()
+    {
+        if (_byteLength == 0)
+        {
+            return BigInteger.Zero;
+        }
+
+        byte[] buffer = ArrayPool<byte>.Shared.Rent(_byteLength);
+        try
+        {
+            BigInteger value;
+            do
+            {
+                RandomNumberGenerator.Fill(buffer.AsSpan(0, _byteLength));
+                if (_bitRemainder != 0)
+                {
+                    buffer[_byteLength - 1] &= _maskByte;
+                }
+
+                value = new BigInteger(buffer, isUnsigned: true, isBigEndian: false);
+            }
+            while (value >= _modulus);
+
+            return value;
+        }
+        finally
+        {
+            ArrayPool<byte>.Shared.Return(buffer, clearArray: true);
+        }
+    }
+
+    private static ulong AbsSub64(ulong a, ulong b)
+    {
+        return a >= b ? a - b : b - a;
+    }
+
+    private static UInt128 AbsSub128(UInt128 a, UInt128 b)
+    {
+        return a >= b ? a - b : b - a;
+    }
+
+    private static BigInteger AbsSubBigInteger(BigInteger a, BigInteger b)
+    {
+        BigInteger diff = a - b;
+        return diff.Sign < 0 ? -diff : diff;
+    }
+
+    private static ulong BinaryGcd64(ulong u, ulong v)
+    {
+        if (u == 0UL)
+        {
+            return v;
+        }
+
+        if (v == 0UL)
+        {
+            return u;
+        }
+
+        int shift = BitOperations.TrailingZeroCount(u | v);
+        u >>= BitOperations.TrailingZeroCount(u);
+        do
+        {
+            v >>= BitOperations.TrailingZeroCount(v);
+            if (u > v)
+            {
+                (u, v) = (v, u);
+            }
+
+            v -= u;
+        }
+        while (v != 0UL);
+
+        return u << shift;
+    }
+
+    private static void MultiplyFull(UInt128 left, UInt128 right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
+    {
+        ulong leftLow = (ulong)left;
+        ulong leftHigh = (ulong)(left >> 64);
+        ulong rightLow = (ulong)right;
+        ulong rightHigh = (ulong)(right >> 64);
+
+        (ulong High, ulong Low) partial0 = Multiply64(leftLow, rightLow);
+        (ulong High, ulong Low) partial1 = Multiply64(leftLow, rightHigh);
+        (ulong High, ulong Low) partial2 = Multiply64(leftHigh, rightLow);
+        (ulong High, ulong Low) partial3 = Multiply64(leftHigh, rightHigh);
+
+        p0 = partial0.Low;
+        ulong carry = partial0.High;
+        ulong sum = carry + partial1.Low;
+        ulong carryMid = sum < partial1.Low ? 1UL : 0UL;
+        sum += partial2.Low;
+        if (sum < partial2.Low)
+        {
+            carryMid++;
+        }
+
+        p1 = sum;
+        sum = partial1.High + partial2.High;
+        ulong carryHigh = sum < partial2.High ? 1UL : 0UL;
+        sum += partial3.Low;
+        if (sum < partial3.Low)
+        {
+            carryHigh++;
+        }
+
+        sum += carryMid;
+        if (sum < carryMid)
+        {
+            carryHigh++;
+        }
+
+        p2 = sum;
+        p3 = partial3.High + carryHigh;
+    }
+
+    private static (ulong High, ulong Low) Multiply64(ulong left, ulong right)
+    {
+        ulong a0 = (uint)left;
+        ulong a1 = left >> 32;
+        ulong b0 = (uint)right;
+        ulong b1 = right >> 32;
+
+        ulong lo = a0 * b0;
+        ulong mid1 = a1 * b0;
+        b0 = a0 * b1;
+        b1 *= a1;
+
+        a0 = (lo >> 32) + (uint)mid1 + (uint)b0;
+        a1 = (lo & 0xFFFFFFFFUL) | (a0 << 32);
+        b1 += (mid1 >> 32) + (b0 >> 32) + (a0 >> 32);
+
+        return (b1, a1);
+    }
+
+    private static UInt128 ReduceProductBitwise(ulong p3, ulong p2, ulong p1, ulong p0, UInt128 modulus)
+    {
+        UInt128 remainder = UInt128.Zero;
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p3 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p2 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p1 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        for (int bit = 63; bit >= 0; bit--)
+        {
+            remainder <<= 1;
+            remainder |= (UInt128)((p0 >> bit) & 1UL);
+            if (remainder >= modulus)
+            {
+                remainder -= modulus;
+            }
+        }
+
+        return remainder;
+    }
+
+    private static UInt128 ReadUInt128LittleEndian(ReadOnlySpan<byte> buffer)
+    {
+        UInt128 value = UInt128.Zero;
+        for (int i = buffer.Length - 1; i >= 0; i--)
+        {
+            value <<= 8;
+            value |= buffer[i];
+        }
+
+        return value;
+    }
+}

PerfectNumbers.Core/Gpu/PrimeOrderGpuHeuristics.cs

@@ -4,12 +4,10 @@ using System.Collections.Concurrent;
 using ILGPU;
 using ILGPU.Runtime;
 using System.Numerics;
-using System.Globalization;
 using PerfectNumbers.Core;
 using System.Runtime.InteropServices;
 using System.Runtime.CompilerServices;
 using ILGPU.Runtime.OpenCL;
-
 namespace PerfectNumbers.Core.Gpu;
 
 internal enum GpuPow2ModStatus
@@ -29,25 +27,6 @@ internal static partial class PrimeOrderGpuHeuristics
     private static readonly ConcurrentDictionary<Accelerator, OrderKernelLauncher> OrderKernelCache = new();
     private static readonly ConcurrentDictionary<Accelerator, SmallPrimeDeviceCache> SmallPrimeDeviceCaches = new();
 
-    private static void ReportPow2FailureCore(string context, string modulusDescription, GpuPow2ModStatus status)
-    {
-        string message = $"GPU pow2mod failed with status {status} for divisor {modulusDescription} during {context}.";
-        Console.Error.WriteLine(message);
-        throw new InvalidOperationException(message);
-    }
-
-    internal static void ReportPow2Failure(string context, ulong modulus, GpuPow2ModStatus status)
-    {
-        string modulusDescription = modulus.ToString(CultureInfo.InvariantCulture);
-        ReportPow2FailureCore(context, modulusDescription, status);
-    }
-
-    internal static void ReportPow2Failure(string context, UInt128 modulus, GpuPow2ModStatus status)
-    {
-        string modulusDescription = modulus.ToString(CultureInfo.InvariantCulture);
-        ReportPow2FailureCore(context, modulusDescription, status);
-    }
-
     public readonly struct OrderKernelConfig(ulong previousOrder, byte hasPreviousOrder, uint smallFactorLimit, int maxPowChecks, int mode)
     {
         public readonly ulong PreviousOrder = previousOrder;

PerfectNumbers.Core/MersenneDivisorCycles.cs

@@ -265,12 +265,11 @@ public class MersenneDivisorCycles
     }
 
     public static bool TryCalculateCycleLengthForExponent(
-        ulong divisor,
+        ulong divisor,        
         ulong exponent,
         in MontgomeryDivisorData divisorData,
         Dictionary<ulong, FactorCacheEntry>? factorCache,
-        out ulong cycleLength,
-        bool allowGpuPow2 = false)
+        out ulong cycleLength)
     {
         cycleLength = 0UL;
 
@@ -291,8 +290,7 @@ public class MersenneDivisorCycles
             previousOrder: null,
             divisorData,
             PrimeOrderCalculator.PrimeOrderSearchConfig.HeuristicDefault,
-            PrimeOrderCalculator.PrimeOrderHeuristicDevice.Cpu,
-            allowGpuPow2);
+            PrimeOrderCalculator.PrimeOrderHeuristicDevice.Cpu);
         if (orderResult.Order != 0UL)
         {
             cycleLength = orderResult.Order;
@@ -329,7 +327,7 @@ public class MersenneDivisorCycles
             return false;
         }
 
-        cycleLength = ReduceOrder(divisorData, phi, factorCounts, allowGpuPow2);
+        cycleLength = ReduceOrder(divisorData, phi, factorCounts);
         return true;
     }
 
@@ -473,7 +471,7 @@ public class MersenneDivisorCycles
         }
     }
 
-    private static ulong ReduceOrder(in MontgomeryDivisorData divisorData, ulong initialOrder, Dictionary<ulong, int> factorCounts, bool allowGpuPow2)
+    private static ulong ReduceOrder(in MontgomeryDivisorData divisorData, ulong initialOrder, Dictionary<ulong, int> factorCounts)
     {
         if (factorCounts.Count == 0)
         {
@@ -507,7 +505,7 @@ public class MersenneDivisorCycles
                     }
 
                     ulong candidate = order / prime;
-                    if (Pow2EqualsOne(candidate, divisorData, allowGpuPow2))
+                    if (candidate.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL)
                     {
                         order = candidate;
                         continue;
@@ -525,26 +523,6 @@ public class MersenneDivisorCycles
         }
     }
 
-    [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private static bool Pow2EqualsOne(ulong exponent, in MontgomeryDivisorData divisorData, bool allowGpuPow2)
-    {
-        if (allowGpuPow2)
-        {
-            GpuPow2ModStatus status = PrimeOrderGpuHeuristics.TryPow2Mod(exponent, divisorData.Modulus, out ulong remainder, divisorData);
-            if (status == GpuPow2ModStatus.Success)
-            {
-                return remainder == 1UL;
-            }
-
-            PrimeOrderGpuHeuristics.ReportPow2Failure(
-                "divisor cycle reduction",
-                divisorData.Modulus,
-                status);
-        }
-
-        return exponent.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL;
-    }
-
     private static ulong PollardRho64(ulong n)
     {
         if ((n & 1UL) == 0UL)
@@ -946,7 +924,7 @@ public class MersenneDivisorCycles
         return CalculateCycleLengthFallback(divisor);
     }
 
-    internal static bool TryCalculateCycleLengthHeuristic(ulong divisor, in MontgomeryDivisorData divisorData, out ulong cycleLength, bool allowGpuPow2 = false)
+    internal static bool TryCalculateCycleLengthHeuristic(ulong divisor, in MontgomeryDivisorData divisorData, out ulong cycleLength)
     {
         if ((divisor & (divisor - 1UL)) == 0UL)
         {
@@ -967,8 +945,7 @@ public class MersenneDivisorCycles
                     previousOrder: null,
                     divisorData,
                     PrimeOrderCalculator.PrimeOrderSearchConfig.HeuristicDefault,
-                    PrimeOrderCalculator.PrimeOrderHeuristicDevice.Cpu,
-                    allowGpuPow2);
+                    PrimeOrderCalculator.PrimeOrderHeuristicDevice.Cpu);
             if (orderResult.Order != 0UL)
             {
                 cycleLength = orderResult.Order;

PerfectNumbers.Core/PrimeOrderCalculator.Wide.cs

@@ -31,8 +31,7 @@ internal static partial class PrimeOrderCalculator
         PrimeOrderSearchConfig config,
         PrimeOrderHeuristicDevice device)
     {
-        bool enableGpuPow2 = device == PrimeOrderHeuristicDevice.Gpu;
-        using var scope = UsePow2Mode(device, enableGpuPow2);
+        using var scope = UsePow2Mode(device);
         MontgomeryDivisorData divisorData;
         if (prime <= ulong.MaxValue)
         {
@@ -52,7 +51,7 @@ internal static partial class PrimeOrderCalculator
 
             ulong prime64 = (ulong)prime;
             divisorData = MontgomeryDivisorData.FromModulus(prime64);
-            PrimeOrderResult result = Calculate(prime64, previous, divisorData, config, device, enableGpuPow2);
+            PrimeOrderResult result = Calculate(prime64, previous, divisorData, config, device);
             return new PrimeOrderResultWide(result.Status, result.Order);
         }
         else
@@ -1031,11 +1030,6 @@ internal static partial class PrimeOrderCalculator
             {
                 return wideRemainder;
             }
-
-            PrimeOrderGpuHeuristics.ReportPow2Failure(
-                "wide prime order pow2 check",
-                modulus,
-                wideStatus);
         }
 
         return exponent.Pow2MontgomeryModWindowed(modulus);

PerfectNumbers.Core/PrimeOrderCalculator.cs

@@ -90,13 +90,13 @@ internal static partial class PrimeOrderCalculator
         private readonly bool _previousAllow;
         private readonly PrimeOrderHeuristicDevice _previousDevice;
 
-        public Pow2ModeScope(PrimeOrderHeuristicDevice device, bool allowGpuPow2)
+        public Pow2ModeScope(PrimeOrderHeuristicDevice device)
         {
             _previousInitialized = s_pow2ModeInitialized;
             _previousAllow = s_allowGpuPow2;
             _previousDevice = s_deviceMode;
             s_pow2ModeInitialized = true;
-            s_allowGpuPow2 = allowGpuPow2;
+            s_allowGpuPow2 = device == PrimeOrderHeuristicDevice.Gpu;
             s_deviceMode = device;
         }
 
@@ -124,7 +124,7 @@ internal static partial class PrimeOrderCalculator
         }
     }
 
-    private static Pow2ModeScope UsePow2Mode(PrimeOrderHeuristicDevice device, bool allowGpuPow2) => new(device, allowGpuPow2);
+    private static Pow2ModeScope UsePow2Mode(PrimeOrderHeuristicDevice device) => new(device);
 
     private static DebugLoggingScope UseDebugLogging(bool enabled) => new(enabled);
 
@@ -154,11 +154,9 @@ internal static partial class PrimeOrderCalculator
         ulong? previousOrder,
         in MontgomeryDivisorData divisorData,
         PrimeOrderSearchConfig config,
-        PrimeOrderHeuristicDevice device,
-        bool allowGpuPow2 = false)
+        PrimeOrderHeuristicDevice device)
     {
-        bool enableGpuPow2 = allowGpuPow2 || device == PrimeOrderHeuristicDevice.Gpu;
-        using var scope = UsePow2Mode(device, enableGpuPow2);
+        using var scope = UsePow2Mode(device);
         return CalculateInternal(prime, previousOrder, divisorData, config);
     }
 
@@ -445,6 +443,7 @@ internal static partial class PrimeOrderCalculator
         int powBudget = config.MaxPowChecks <= 0 ? candidates.Count : config.MaxPowChecks;
         int powUsed = 0;
         int candidateCount = candidates.Count;
+        bool allowGpuBatch = true;
         Span<ulong> candidateSpan = CollectionsMarshal.AsSpan(candidates);
 
         DebugLog(() => $"Checking candidates ({candidateCount} candidates, {powBudget} pow budget)");
@@ -468,11 +467,9 @@ internal static partial class PrimeOrderCalculator
             bool gpuSuccess = false;
             bool gpuStackRemainders = false;
             GpuPow2ModStatus status = GpuPow2ModStatus.Unavailable;
-            bool attemptedGpu = false;
 
-            if (IsGpuPow2Allowed)
+            if (allowGpuBatch && IsGpuPow2Allowed)
             {
-                attemptedGpu = true;
                 if (batchSize <= StackGpuBatchSize)
                 {
                     Span<ulong> localRemainders = stackGpuRemainders.Slice(0, batchSize);
@@ -499,14 +496,11 @@ internal static partial class PrimeOrderCalculator
                         gpuPool = null;
                     }
                 }
-            }
 
-            if (attemptedGpu && !gpuSuccess)
-            {
-                PrimeOrderGpuHeuristics.ReportPow2Failure(
-                    "prime order candidate scan",
-                    prime,
-                    status);
+                if (!gpuSuccess && (status == GpuPow2ModStatus.Overflow || status == GpuPow2ModStatus.Unavailable))
+                {
+                    allowGpuBatch = false;
+                }
             }
 
             for (int i = 0; i < batchSize && powUsed < powBudget; i++)
@@ -786,11 +780,6 @@ internal static partial class PrimeOrderCalculator
             {
                 return remainder == 1UL;
             }
-
-            PrimeOrderGpuHeuristics.ReportPow2Failure(
-                "prime order pow2 check",
-                prime,
-                status);
         }
 
         return exponent.Pow2MontgomeryModWindowed(divisorData, keepMontgomery: false) == 1UL;