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Friday, June 06, 2014

Some cubic modular work

As much as I talk about binary quadratic forms turns out I also have extended concepts pioneered with them to cubic residues.

So I have found the following modular solution.

Given:  x3 - Dy3 = F

I can solve modulo N, where N is a cubic residue of D, m is the cubed residue, and r is a residue modulo N, where its modular inverse exists:

x = my + r mod N

and

3(2my+r)2  = 4Fr-1 - r2  mod N

If you wish to see it used, you can just check:

somemath.blogspot.com/2012/05/cubic-diophantine-computations.html

The ideas I use can be generalized further. But I think I went just to a cubic just to see what it might look like. Result seems very pure to me as in I'm not sure how useful it is.


James Harris
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