factor with method selection

[s-celles]

Hello,

Currently, factor() uses a hardcoded algorithm chain: trial division up to 2^32, then pollardfactor(). With ECM and eventually MPQS being added, users would benefit from:

1. Explicitly choosing a factorization algorithm
2. Tuning algorithm parameters (e.g., ECM smoothness bound)
3. A clean extension point for future algorithms

Proposed API

Following SciML conventions (cf. solve(prob, alg) in DifferentialEquations.jl ...):

 abstract type FactorizationMethod end
 struct TrialDivision <: FactorizationMethod end
 struct PollardRho    <: FactorizationMethod end
 struct ECM           <: FactorizationMethod
     B1::Int
     num_curves::Int
 end
 struct Auto          <: FactorizationMethod end
 
 # Keyword API (backwards compatible)
 factor(n; method=Auto())
 
 # Positional dispatch (Julian style)
 factor(n, ::TrialDivision)
 factor(n, ::PollardRho)
 factor(n, ::ECM)
 factor(n, ::Auto)          # current polyalgorithm heuristic

Auto() would encode the current behavior (trial division → Pollard rho) and grow to include ECM/MPQS size-based heuristics.

Backward compatibility

• factor(n) continues to work unchanged (defaults to Auto())
• factor(ContainerType, n) unaffected
• eachfactor(n) would also accept a method keyword

Considerations

• Should strategy types be exported? Probably yes for discoverability.
• Should eachfactor support methods too, or only factor?
• Parameterless strategies (TrialDivision(), PollardRho()) could carry default tuning parameters later without breaking API.

Related:

• Factorization of "large" numbers #159 (comment)