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COMMENTS
| Representing a given prime P=3k+1 as x^2+3y^2 amounts to finding the shortest vector in a 2-dimensional lattice, namely either of the primes above P in the ring Q(sqrt(-3)). For instance, if P = 2^521 - 1 then P = x^2 + 3y^2 where x,y are 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 898670952308059000662208200339860406351380028634597445743368513219427297854627. - Noam D. Elkies, Jun 25, 2001
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PROG
| (PARI) f(p, P, a, m)= P=2^p-1; a=lift(sqrt(Mod(-3, P))); m=[P, a; 0, 1]; (m*qflll(m, 1))~[1, ]
for(n=1, 11, print(abs(f([3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 521][n])[1])))
(PARI) f(p, P, a, m)= P=2^p-1; a=lift(sqrt(Mod(-3, P))); m=[P, a; 0, 1]; (m*qflll(m, 1))~[1, ] for(n=1, 12, print(abs(f([3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521][n])[1]))) - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 23 2006
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