cp's OEIS Frontend

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.

%I A351026 #25 Feb 16 2025 08:34:02
%S A351026 51,85,102,119,170,204,238,291,340,408,459,476,485,579,582,663,679,
%T A351026 680,697,723,765,771,816,867,918,952,965,970,1071,1105,1158,1164,1205,
%U A351026 1275,1285,1326,1351,1358,1360,1394,1445,1446,1530,1542,1547,1632,1687,1734,1785
%N 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.
%H A351026 R. D. Carmichael, <a href="http://www.jstor.org/stable/2971748">Fermat numbers F(n) = 2^(2^n) + 1</a>, Amer. J. Math., 26 (1919), 137-146.
%H A351026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PepinsTest.html">Pepin's Test</a>
%F A351026 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.
%o A351026 (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, ", ")))));
%Y A351026 Cf. A028730, A129802, A136804, A136806.
%K A351026 nonn
%O A351026 1,1
%A A351026 _Arkadiusz Wesolowski_, Jan 29 2022