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I misinterpreted "Other reductions possible" to read "Further reductions possible" given the example and where the commentary appeared. Admittedly I haven't read the entire thread and was under the incorrect impression for any given input there was only 1 acceptable output. I was also under the incorrect impression that you were not interested in prime factorization but what would be the result if the problem of "cancelling out" had been attacked by hand.
I believe the approach I presented could be made more efficient though I doubt it would be able to compete with a good prime factorization algorithm. I am familiar with several, some may have even been used in this thread, such as the quadratic sieve - but I don't currently have the time to write a perl implementation. IIRC numbers smaller than a certain size are best done with 1 algorithm and larger than a different number with a different algorithm with the numbers falling in the middle being a toss up.
If I do come up with anything I will be sure to let you know. Thanks for the fun problem.
Cheers - L~R
In reply to Re^3: Algorithm for cancelling common factors between two lists of multiplicands