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A123271
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Sign of the penultimate term of the Lucas-Lehmer sequence modulo the n-th Mersenne prime.
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0
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1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1, -1, 1
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OFFSET
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2,1
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COMMENTS
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For the n-th Mersenne prime 2^p - 1 = A000668(n) (with p=A000043(n)), we have A003010(p-2) == 0 (mod 2^p - 1). Therefore A003010(p-3) == a(n) * 2^((p+1)/2) (mod 2^p - 1) where a(n) = 1 or -1.
From currently known Mersenne primes with exponents 57885161, 74207281, 77232917, 82589933 we have the sequence values (-1, -1, 1, -1), but there is a possibility of new Mersenne primes to be found out of order. - Serge Batalov, Feb 04 2013; updated by Max Alekseyev, Feb 25 2018, updated by Gord Palameta, Dec 21 2018
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LINKS
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Table of n, a(n) for n=2..47.
Bastiaan Jansen, Mersenne primes and class field theory. Doctoral thesis, Leiden University, 2012.
Mersenne Forum, Penultimate Lucas-Lehmer step
Eric Weisstein's World of Mathematics, Lucas-Lehmer test
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FORMULA
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a(n) = 1 or -1 such that A003010(A000043(n)-3) = a(n) * 2^((A000043(n)+1)/2) (mod A000668(n)).
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EXAMPLE
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From Serge Batalov, Feb 04 2013: (Start)
For n=3, p=5, M_p=31, and the Lucas-Lehmer sequence is (4, 14, 8, 0). The penultimate element is 1*2^3 mod M_p = 8 mod 31, so a(3)=1.
For n=4, p=7, M_p=127, and the Lucas-Lehmer sequence is (4, 14, 67, 42, 111, 0). The penultimate element is -1*2^4 mod M_p = 111 mod 127, so a(4)=-1.
(End)
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PROG
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(PARI) test(p)=s=Mod(4, 2^p-1); for(i=1, p-3, s=s^2-2); r=2^((p+1)/2); if(s==+r, +1, s==-r, -1, "error") \\ Then a(n) = test(A000043(n)). From Jeppe Stig Nielsen, Jan 25 2016
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CROSSREFS
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Cf. A000043, A000668, A003010.
Sequence in context: A015772 A016288 A016030 * A127252 A121238 A321753
Adjacent sequences: A123268 A123269 A123270 * A123272 A123273 A123274
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KEYWORD
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more,sign
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AUTHOR
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Max Alekseyev, Oct 10 2006, Sep 29 2007
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EXTENSIONS
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More terms from Andreas Höglund, Sep 29 2007
a(40) added by Max Alekseyev, Feb 07 2011
a(41)-a(46) and prospective a(47)-a(48) from Andreas Höglund via Serge Batalov, Feb 04 2013; Max Alekseyev, Feb 25 2018
a(47) confirmed and prospective a(49)-a(51) from Gord Palameta, Dec 21 2018
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STATUS
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approved
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