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

1,1

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

References

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

Links

Table of n, a(n) for n=1..13.

Phil Moore, Tony Reix and others, Online Discussion

Example

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

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

Crossrefs

Cf. A000043, A000668, A059495.

Sequence in context: A001885 A300641 A078433 * A052647 A326983 A232974

Adjacent sequences: A059491 A059492 A059493 * A059495 A059496 A059497

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 05 2001

Extensions

More terms from Noam D. Elkies, Jun 25 2001

Corrected and extended by Joshua Zucker, May 23 2006

Status

approved