Here's a benchmark for Math::Big::Factors wheels of orders 5-7:

    Rate      7      6      5
7 34.1/s     --   -95%  -100%
6  627/s  1739%     --   -94%
5 9770/s 28548%  1458%     --
[download]

and for factors_wheel of numbers near 32 bits, using an order 7 wheel:

           s/iter 4294967291 4294967293 4294967295
4294967291   4.45         --       -61%       -71%
4294967293   1.74       156%         --       -25%
4294967295   1.31       241%        33%         --
[download]

Note that 4294967291 is prime, 4294967293 has 2 large factors, and 4294967295 has 5 factors, the largest of which is 2**16+1.

For 4294967291, and wheel orders 5-7:

  s/iter    7    5    6
7   4.44   -- -16% -21%
5   3.71  19%   --  -6%
6   3.49  27%   6%   --
[download]

In all fairness, I should probably mention that I've customized my own version of Math::Big::Factors to Memoize results where possible, and to reduce the calls to Math::BigInt::new() for constants.

Note that factors_wheel creates a new wheel every time (what a shame), instead of requiring a wheel reference be passed in. Caching wheel goes a little way in correcting this, without changing the interface.

You might also consider avoiding the use of Math::BigInt where possible, as you don't need numbers that big.

Now, back to the question at hand. For numbers near 2**32, factoring a prime number seems to take 150 times longer than creating the order 7 wheel. Creating the order 6 wheel is considerably faster, and so is the factoring based on that wheel.

I'm not sure where the breakpoint is for order 7 wheels. A casual search hints that it is very large.

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