A059494 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.

2, 2, 10, 46, 362, 298, 46162, 1505304098, 17376907720394, 9286834445316902, 9328321181472828398, 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 22958222111004899714849436789827362390710508069726899926224050897274623732073762499062593658

Offset: 1

Author

N. J. A. Sloane, Feb 05 2001

Keywords

• nonn
• easy

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

Examples

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

References

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

Links

• Phil Moore, Tony Reix and others, Online Discussion

Crossrefs

Cf. A000043, A000668, A059495.

Programs

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]))) \\ Joshua Zucker, May 23 2006

Extensions

More terms from Noam D. Elkies, Jun 25 2001

Corrected and extended by Joshua Zucker, May 23 2006