A123271 Sign of the penultimate term of the Lucas-Lehmer sequence modulo the n-th Mersenne prime.

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

Offset

2,1

Comments

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

Links

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

Formula

a(n) = 1 or -1 such that A003010(A000043(n)-3) = a(n) * 2^((A000043(n)+1)/2) (mod A000668(n)).

Example

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)

Prog

(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

Crossrefs

Cf. A000043, A000668, A003010.

Sequence in context: A015772 A016288 A016030 * A121238 A321753 A186032

Adjacent sequences: A123268 A123269 A123270 * A123272 A123273 A123274

Keyword

more,sign

Author

Max Alekseyev, Oct 10 2006, Sep 29 2007

Extensions

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

Status

approved