1,1

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

F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 59.

Phil Moore, Tony Reix and others, Online Discussion

More terms from a Lycos page in Google's cache (original page seems no longer to exist and not to be in the internet archive)

p=7: 127 = 10^2 + 3*3^2, so a(3) = 10.

(PARI/GP) 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

Cf. A000043, A000668, A059495.

Sequence in context: A151364 A001885 A078433 this_sequence A052647 A032034 A002250

Adjacent sequences: A059491 A059492 A059493 this_sequence A059495 A059496 A059497

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Feb 05 2001

More terms from Noam D. Elkies, Jun 25, 2001

Corrected and extended by Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 23 2006