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A059494
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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.
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1
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2, 2, 10, 46, 362, 298, 46162, 1505304098, 17376907720394, 9286834445316902, 9328321181472828398, 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 22958222111004899714849436789827362390710508069726899926224050897274623732073762499062593658
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 59.
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LINKS
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EXAMPLE
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p=7: 127 = 10^2 + 3*3^2, so a(3) = 10.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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