

A013998


Unrestricted Perrin pseudoprimes.


8



271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291, 102690901, 130944133, 196075949, 214038533, 517697641, 545670533, 801123451, 855073301, 903136901, 970355431
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

"The column Mathematical Recreations by Ian Stewart in the June 1996 issue of Scientific American discusses the Perrin sequence [A001608] A(n) defined by A(0)=3, A(1)=0, A(2)=2, A(n+1)=A(n1)+A(n2). Motivated by a theorem of E. Lucas: If n is prime it divides A(n) exactly, the question whether primality of n follows from n divides A(n) exactly was formulated 1899. So far, they say, nobody has found a composite n that divides A(n). Such a number would be called a Perrin pseudoprime. The article quotes an experiment by Steven Arno of the Supercomputing Research Center in Bowie, Md., where a lower bound of 15 digits for the size of the smallest Perrin pseudoprime was obtained in 1991. On Jul 3rd, 1996, I was able to find the two smallest Perrin pseudoprimes:"  Holzbaur
In the "Feedback" section of his column for November 1996, Ian Stewart mentions that Jeffrey Shallit (Waterloo) had written to him saying that he had found the Perrin pseudoprimes 271441 and 904631 in 1982.


REFERENCES

W. W. Adams and D. Shanks, Strong primality tests that are not sufficient, Math. Comp. 39 (1982), 255300.
Ian Stewart, Tales of a neglected number, Scientific American, No. 6, 1966, pp. 9293.


LINKS

Robert Harley, Table of n, a(n) for n = 1..658
Christian Holzbaur, Perrin pseudoprimes, [broken link, see U Vienna personnel ]
Ian Stewart, "Tales of a Neglected Number" [broken link]
Eric Weisstein's World of Mathematics, Perrin Pseudoprime.
Index entries for sequences related to pseudoprimes


PROG

(PARI)
N=10^10;
default(primelimit, N);
M = [0, 1, 0; 0, 0, 1; 1, 1, 0];
a(n)=lift( trace( Mod(M, n)^n ) ); /* A215339(n) */
{ for (n=1, N,
if ( isprime(n), next() );
if ( a(n)==0, print1(n, ", "); );
); }
/* Joerg Arndt_, Aug 16 2012 */


CROSSREFS

Cf. A018187.
Sequence in context: A244561 A137715 A237181 * A236623 A214193 A232256
Adjacent sequences: A013995 A013996 A013997 * A013999 A014000 A014001


KEYWORD

nonn,more


AUTHOR

R. K. Guy


EXTENSIONS

More terms from alipson(AT)cix.compulink.co.uk (Andrew Lipson). Further terms beyond those shown here have been computed by cdw10(AT)cix.compulink.co.uk (C Wright).
Holzbaur quote from Robert G. Wilson, Nov 30 2001


STATUS

approved



