A129802 Possible bases for Pepin's primality test for Fermat numbers.

3, 5, 6, 7, 10, 12, 14, 20, 24, 27, 28, 39, 40, 41, 45, 48, 51, 54, 56, 63, 65, 75, 78, 80, 82, 85, 90, 91, 96, 102, 105, 108, 112, 119, 125, 126, 130, 147, 150, 156, 160, 164, 170, 175, 180, 182, 192, 204, 210, 216, 224, 238, 243, 245, 250, 252, 260, 291, 294, 300

Offset

1,1

Comments

Prime elements of this sequence are given by A102742.

Links

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pepin's Test

Formula

A positive integer 2^k*m, where m is odd and k>=0, belongs to this sequence iff the Jacobi symbol (F_n/m)=1 only for a finite number of Fermat numbers F_n=A000215(n).

Prog

(PARI) { isPepin(n) = local(s, S=Set(), t); n\=2^valuation(n, 2); s=Mod(3, n); while( !setsearch(S, s), S=setunion(S, [s]); s=(s-1)^2+1); t=s; until( t==s, if( kronecker(lift(t), n)==1, return(0)); t=(t-1)^2+1); 1 }
for(n=2, 1000, if(isPepin(n), print1(n, ", ")))
(PARI) for(b=2, 300, k=b/2^valuation(b, 2); if(k>1, i=logint(k, 2); m=Mod(2, k); z=znorder(m); e=znorder(Mod(2, z/2^valuation(z, 2))); t=0; for(c=1, e, if(kronecker(lift(m^2^(i+c))+1, k)==-1, t++, break)); if(t==e, print1(b, ", ")))); \\ Arkadiusz Wesolowski, Sep 22 2021

Crossrefs

Cf. A000215, A019434, A060377, A102742.

Sequence in context: A034035 A335911 A136804 * A023854 A324511 A360008

Adjacent sequences: A129799 A129800 A129801 * A129803 A129804 A129805

Keyword

nonn

Author

Max Alekseyev, Jun 14 2007, corrected Dec 29 2007. Thanks to Ant King for pointing out an error in the earlier version of this sequence.

Status

approved