A351026 Possible bases b > 17 which can be used in Pepin's test to check the primality of any Fermat number greater than 5 only in the case when the base b is smaller than the tested number.

51, 85, 102, 119, 170, 204, 238, 291, 340, 408, 459, 476, 485, 579, 582, 663, 679, 680, 697, 723, 765, 771, 816, 867, 918, 952, 965, 970, 1071, 1105, 1158, 1164, 1205, 1275, 1285, 1326, 1351, 1358, 1360, 1394, 1445, 1446, 1530, 1542, 1547, 1632, 1687, 1734, 1785

Offset

1,1

Links

Table of n, a(n) for n=1..49.

R. D. Carmichael, Fermat numbers F(n) = 2^(2^n) + 1, Amer. J. Math., 26 (1919), 137-146.

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

Formula

A positive integer b belongs to this sequence if and only if the Jacobi symbol J(b,F(m)) has value 0 or 1 for some 5 < F(m) < b, and J(b,F(m)) = 1 only for a finite number of Fermat numbers F(m) = 2^(2^m) + 1.

Prog

(PARI) for(b=18, 1785, a=q=0; until(b-2<16^(2^a), a++; if(!(kronecker(b, 16^(2^(a-1))+1)==-1), q=1; break)); if(q==1, 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, ", ")))));

Crossrefs

Cf. A028730, A129802, A136804, A136806.

Sequence in context: A050698 A039474 A253020 * A020180 A049328 A075894

Adjacent sequences: A351023 A351024 A351025 * A351027 A351028 A351029

Keyword

nonn

Author

Arkadiusz Wesolowski, Jan 29 2022

Status

approved