The Integer Sequence defined by the recurrence

(1) |

(2) |

(3) | |||

(4) | |||

(5) |

The solution is then

(6) |

(7) |

Perrin (1899) investigated the sequence and noticed that if is Prime, then . The first statement of this fact is attributed to É. Lucas in 1876 by Stewart (1996). Perrin also searched for but did not find any Composite Number in the sequence such that . Such numbers are now known as Perrin Pseudoprimes. Malo (1900), Escot (1901), and Jarden (1966) subsequently investigated the series and also found no Perrin Pseudoprimes. Adams and Shanks (1982) subsequently found that 271,441 is such a number.

**References**

Adams, W. and Shanks, D. ``Strong Primality Tests that Are Not Sufficient.'' *Math. Comput.* **39**, 255-300, 1982.

Escot, E.-B. ``Solution to Item 1484.'' *L'Intermédiare des Math.* **8**, 63-64, 1901.

Jarden, D. *Recurring Sequences.* Jerusalem: Riveon Lematematika, 1966.

Perrin, R. ``Item 1484.'' *L'Intermédiare des Math.* **6**, 76-77, 1899.

Stewart, I. ``Tales of a Neglected Number.'' *Sci. Amer.* **274**, 102-103, June 1996.

Sloane, N. J. A. Sequence
A001608/M0429
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26