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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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
Sequence in context: A050698 A039474 A253020 * A020180 A049328 A075894
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified July 22 23:30 EDT 2024. Contains 374544 sequences. (Running on oeis4.)