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Old 2009-08-25, 07:32   #1
Jul 2009

31 Posts
Default Breaking a prime p into a^2 + 3* b^2

I'm playing with cubic reciprocity formulas.

From that link, it states "A theorem of Fermat states that every prime p ≡ 1 (mod 3) is the sum of a square and three times a square: p = a^2 + 3b^2"

How would you go about finding a and b given p?

Is there a better strategy than brute force, iterating b=1, b=2, b=3, b=4.. etc and seeing if the remainder (p-3*b^2) is a square?
There are efficient square-detecting routines, but even if they're cheap, you could be iterating up to sqrt(p) times!

How about the case where p=a^2 +2* b^2, which comes up when dealing with quartic reciprocity?

Last fiddled with by SPWorley on 2009-08-25 at 07:32
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