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Old 2010-04-02, 02:42   #11
gd_barnes's Avatar
May 2007
Kansas; USA

236778 Posts


I have been able to generalize these "old" style algebraic factors, not only across all bases, but across all factors.

I now see the pattern of why some prime factors can combine with partial algebraic factors to make a full covering set and some cannot. Although I do not have a formal proof, after testing all prime factors up to 100 and observing several already-found factors > 100, I'm fairly confident in the finding. Here it is:

For a numeric factor (f) on odd n to combine with algebraic factors on even n to make a full covering set, f must be:

f==(1 mod 4)

That's it! Now we can easily add all applicable prime factors and bases to the generallized post here.

What I have not been able to determine is the x and y values in the generalization. The main example for a factor of 5 is:

k=m^2 and m==2 or 3 mod 5.

Where the "2" and "3" are the x and y values and the "5" is the factor f. I do know that x + y must equal f but that's all that I can determine at this point.

Having tested this for all factors < 100, I can now see that 2 more must be added for factors of 73 and 89. I have now formally added them.

Technically, all factors up to 1024 need to be added to the post to cover every possible situation on the project. But it becomes increasingly rare for them to have an impact as f gets higher because as it gets higher, x and y are higher, the bases are higher, and the conjectures become lower. It will be a fairly straight forward task for me to add them up to a factor of 250. So I'll do that this evening.

This effort is a precursor to updating the new bases script to take into account algebraic factors. These "old style" algebraic factorizations will be added first because they appear to cover nearly 90% of all k's that can be eliminated by partial algebraic factors (along with adding the more simple full algebraic factorization where k and base are perfect squares). Later on, the "new" style that will be added where factor f occurs on even n and algebraic factors occur on odd n, which has a somewhat increased level of complexity to narrow down. Both of these only occur on Riesel bases. And then finally, there would be cubes and higher powers, which may never be formally added to the script other than checking for both the k and the base being a perfect cube or higher power. The complexity of cubes and higher powers (if both k and base are not a perfect power) increases exponentially over squares and is far more rare. The time is likely not worth the effort.


Last fiddled with by gd_barnes on 2010-04-02 at 03:13
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