OFFSET
2,1
COMMENTS
Also known as the Lehmer symbol ϵ(4, p) for Mersenne prime exponent p.
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 we have these exponents and sequence values: (74207281: -1, 77232917: 1, 82589933: -1, 136279841: 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, Oct 21 2024
LINKS
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
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 == 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 == 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
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) added by Gord Palameta, Dec 21 2018
a(48) added by Gord Palameta, Oct 21 2024
STATUS
approved