# GPU By-Divisor `MulMod` Usage Audit

This note tracks every `MulMod` call site in the GPU-only `--mersenne=bydivisor` pipeline that changed in the recent optimizer work. It now records where the host path swapped over to the fast helpers and which kernel paths already aligned with the benchmark winners.

## CPU helpers invoked by the GPU path

| Location | Notes |
| --- | --- |
| `ULongExtensions.GpuCompatibleMulModSimplifiedExtension` | Wraps `GpuUInt128.MulModSimplified` so host code mirrors the benchmark winner without reimplementing GPU logic.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L23-L31】 |
| `ULongExtensions.MulMod64Gpu` | Marked obsolete; migrate to `ULongExtensions.GpuCompatibleMulModSimplifiedExtension` for GPU-compatible host code or `GpuUInt128.MulMod` when running inside kernels.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L17-L24】 |
| `ULongExtensions.Pow2ModWindowedGpu` | Migrated to call `GpuCompatibleMulModSimplifiedExtension` for every square and multiply, eliminating the 6–18× slowdown that the shim introduced while executing on the host and aligning host math with the new production helper.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L39-L109】 |
| `ULongExtensions.InitializeStandardOddPowers` | Populates the odd-power table through `GpuCompatibleMulModSimplifiedExtension`, so the host setup costs now match the GPU-compatible benchmark winner.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L286-L309】 |

All host-executed `MulMod` operations in the GPU pipeline now target the optimized CPU helper.

## GPU kernel and device-side code paths

| Location | Notes |
| --- | --- |
| `ULongExtensions.Pow2ModWindowedGpuKernel` | Uses the in-place `GpuUInt128.MulMod` helper, which benchmarks as the fastest device-compatible option when the modulus fits in 64 bits—the case for divisor scans.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L112-L149】【F:Benchmarks.txt†L604-L614】 |
| `ULongExtensions.ComputeWindowedOddPowerGpuKernel` | Also relies on the in-place helper while iterating the odd-power ladder, so no slower variants are in play.【F:PerfectNumbers.Core/ULongExtensions.Gpu.cs†L153-L181】【F:Benchmarks.txt†L604-L614】 |
| `ExponentRemainderStepperGpu.AdvanceStateGpu` | Advances residues via `_currentResidue.MulMod` with the cached modulus. This matches the in-place approach validated by the GPU benchmarks, so nothing to fix here.【F:PerfectNumbers.Core/Gpu/ExponentRemainderStepperGpu.cs†L99-L111】【F:Benchmarks.txt†L604-L614】 |

No other GPU-side `MulMod` calls exist in the by-divisor path, so the device kernels already use the most efficient helper available.