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%I #16 Sep 25 2022 22:55:43
%S 2,2,10,46,362,298,46162,1505304098,17376907720394,9286834445316902,
%T 9328321181472828398,
%U 2107597973657165184339850860393713575649657317180489057212823189967494080057958,22958222111004899714849436789827362390710508069726899926224050897274623732073762499062593658
%N For odd p such that 2^p-1 is a prime (A000043), write 2^p-1 = x^2+3*y^2; sequence gives values of x.
%C 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
%D F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 59.
%H Phil Moore, Tony Reix and others, <a href="http://www.mersenneforum.org/showthread.php?t=345">Online Discussion</a>
%e p=7: 127 = 10^2 + 3*3^2, so a(3) = 10.
%o (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,]
%o for(n=1,11,print(abs(f([3,5,7,13,17,19,31,61,89,107,521][n])[1]))) \\ _Joshua Zucker_, May 23 2006
%Y Cf. A000043, A000668, A059495.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Feb 05 2001
%E More terms from _Noam D. Elkies_, Jun 25 2001
%E Corrected and extended by _Joshua Zucker_, May 23 2006