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 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. 1
 2, 2, 10, 46, 362, 298, 46162, 1505304098, 17376907720394, 9286834445316902, 9328321181472828398, 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 22958222111004899714849436789827362390710508069726899926224050897274623732073762499062593658 (list; graph; refs; listen; history; text; internal format)
 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 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])))) (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])))) \\ 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

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Last modified July 11 03:55 EDT 2020. Contains 335609 sequences. (Running on oeis4.)