20090825, 07:32  #1 
Jul 2009
31 Posts 
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 (p3*b^2) is a square? There are efficient squaredetecting 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 20090825 at 07:32 
20090825, 09:46  #2  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
Factor p over Q(sqrt(3)). See H. Cohen's book on Algebraic Number Theory. I believe that a variation of Cornachia'a algorithm is used, but my memory could be faulty. It's been a long time since I looked at this kind of stuff. 

20090825, 12:08  #3  
Mar 2009
2×19 Posts 
Quote:


20090826, 03:05  #4 
Jul 2009
31 Posts 
Thanks very much for the references.. it's all there in Crandall/Pomerance.
Pretty straightforward, too! I appreciate it. 
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