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Volume Index M ELTHE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION A Report on The Third International Conference on Fibonacci Numbers and Their Applications Herta T. Freitag 289 An Iterated Quadratic Extension of GF(2) Doug Wiedemann 290 A Note on the Primality of 62" + 1 and 102" +1 H.C. Williams 296 Suppose More Rabbits Are Born Shari Lynn Levine 306 New Unitary Perfect Numbers Have at Least Nine Odd Components Charles R. Wall 312 Second International Conference Proceedings 317 A Note on Fibonacci Trees and the Zeckendorf Representation of Integers Renato M. Capocelli 318 A Note on Specially Multiplicative Arithmetic Functions Pentti Haukkanen 325 Indentities Derived on a Fibonacci Multiplication Table Alvin Tirman & T. Henry Jablonski, Jr. 328 On Sums of Three Triangular Numbers John A. Ewell 332 Stroeker's Equation and Fibonacci Numbers. Andrzej Makowski 336 Some Observations on the Classical Cuboid and Its Parametric Solutions W.J.A. Colman 338 Pell Polynomials and a Conjecture of Mahon and Horadam PaulDuvall & Theresa Vaughan 344 Convolution Trees and Pascal-T Triangles John C. Turner 354 A Note on the Third-Order Strong Divisibility Sequences Pavel Horak 366 Elementary Problems and Solutions Edited by A. P. Hillman 372 Advanced Problems and Solutions Edited by Raymond E. Whitney 377 Book Review—77ie Book of Squares A.F. Horadam 382 Volume Index m 7 VVCO LUfVIE 26 NOVEMBER 1988 ISSUSVIBER 4 PURPOSE The primary function of THE FIBONACCI QUARTERLY is to serve as a focal point for widespread interest in the Fibonacci and related numbers, especially with respect to new results, research proposals, challenging problems, and innovative proofs of old ideas. EDITORIAL POLICY THE FIBONACCI QUARTERLY seeks articles that are intelligible yet stimulating to its readers, most of whom are university teachers and students. These articles should be lively and well motivated, with new ideas that develop enthusiasm for number sequences or the explora- tion of number facts. Illustrations and tables should be wisely used to clarify the ideas of the manuscript. Unanswered questions are encouraged, and a complete list of references is abso- lutely necessary. SUBMITTING AN ARTICLE Articles should be submitted in the format of the current issues of the THE FIBONACCI QUARTERLY. They should be typewritten or reproduced typewritten copies, that are clearly readable, double spaced with wide margins and on only one side of the paper. The full name and address of the author must appear at the beginning of the paper directly under the title. Illustra- tions should be carefully drawn in India ink on separate sheets of bond paper or vellum, approx- imately twice the size they are to appear in print. Two copies of the manuscript should be submitted to: GERALD E. BERGUM, EDITOR, THE FIBONACCI QUARTERLY, DEPARTMENT OF COMPUTER SCIENCE, SOUTH DAKOTA STATE UNIVERSITY, BOX 2201, BROOKINGS, SD 57007-0194. Authors are encouraged to keep a copy of their manuscripts for their own files as protection against loss. The editor will give immediate acknowledgment of all manuscripts received. SUBSCRIPTIONS, ADDRESS CHANGE, AND REPRINT INFORMATION Address all subscription correspondence, including notification of address change, to: RICHARD VINE, SUBSCRIPTION MANAGER, THE FIBONACCI ASSOCIATION, SANTA CLARA UNIVERSITY, SANTA CLARA, CA 95053. Requests for reprint permission should be directed to the editor. However, general permis- sion is granted to members of The Fibonacci Association for noncommercial reproduction of a limited quantity of individual articles (in whole or in part) provided complete reference is made to the source. Annual domestic Fibonacci Association membership dues, which include a subscription to THE FIBONACCI QUARTERLY, are $27 for Regular Membership, $37 for Sustain Mem- bership, and $67 for Institutional Membership; foreign rates, which are based on international mailing rates, are somewhat higher than domestic rates; please write for details. THE FIBO- NACCI QUARTERLY is published each February, May, August and November. All back issues of THE FIBONACCI QUARTERLY are available in microfilm or hard copy format from UNIVERSITY MICROFILMS INTERNATIONAL, 300 NORTH ZEEB ROAD, DEPT. P.R., ANN ARBOR, MI 48106. Reprints can also be purchased from UMI CLEARING HOUSE at the same address. 1988 by © The Fibonacci Association All rights reserved, including rights to this journal issue as a whole and, except where otherwise noted, rights to each individual contribution. A Report On The Third International Conference on Fibonacci Numbers And Their Applications Herta T8 Freltag A newspaper article at Pisa, Italy, with a prominent headline: "CONVEGNO PARLANO I MATE- MATICI LTNCONTRO IN OMMAGIO A FIBONACCI" hearalded our Third International Conference on Fibonacci Numbers and Their Applications which was held in Pisa, Italy, July 25th-29th, 1988. A stamp: "I NUMERI DI FIBONACCI CONGRESSO INTERNAZIONALE, 26-7-1988" commemorated it. Of course, mathematicians all across the globe, and especially those who are so fortunate as to have become interested in "Fibonacci-type mathematics," had known about it for some time. The August 1987 issue of The Fibonacci Quarterly had brought the glad tidings: an announcement that our third conference was to take place at the University of Pisa during the last week of July 1988. By mid June 1988, we held the coveted program in our hands. 66 participants were listed, and they came from 22 different countries, the U.S. heading the list with a representation of 20, followed by Italy and Australia. Of course, it was to be expected that at conference time proper additional names would lengthen the count. Forty-five papers were to be presented, several of them with coauthors; there were 3 women speakers. Theoretically sounding titles abounded. There was Andreas N. Philippou's paper, coauthored by Demetris L. Antzoulakes: "Multivariate Fibonacci Polynomials of Order K and the Multiparameter Negative Binomial Distribution of the Same Order." But, rather intriguingly, practical interests wedged themselves in also with Piero Filipponi's paper, coauthored by Emilio Montolivo: "Representation of Natural Numbers as a Sum of Fibonacci Numbers: An Application to Modern Cryptography." This again highlighted one of the joys mathematicians experience: the interplay between theoretical and applied mathematics. What a delight it was to meet in Pisa, Italy, the birthplace of Leonardo of Pisa, son of Bonacci, "our" Fibonacci (= 1170-1250). We already knew that—befittingly, and much to our pleasure—Pisa had honored its mathematical son by a statue. My friends and I were among the many (maybe it was all of them) who made a pilgrimage to Fibonacci's statue. It was a fairly long walk, eventually on Via Fibonacci(l), along the Arno River, until we finally found him in a pretty little park. He seemed thoughtful, and appeared to enjoy the sight of the nearby shrubs and flowers. I felt like thanking him for "having started it all," for having coined the sequence that now bears his name. It would have been nice to invite him to our sessions. I predict he would have been thoroughly startled. What had happened since 1202 when his Liber Abaci was published?! Almost invariably, the papers were of very high caliber. The great variety of topics and the multitude of approaches to deal with a given mathematical idea was remarkable and rather appealing. And it was inspiring to coexperience the deep involvement which authors feel with their topic. We worked hard. The sessions started at 9 a.m. and with short intermissions (coffee break and lunch) they lasted till about 5:30 p.m. As none of the papers were scheduled simultaneously, we could experience the luxury of hearing ALL presentations. We did take out time to play. Of course, just to BE in Pisa was a treat. We stepped into the past, enwrapped into the charm of quaint, old buildings, which—could they only talk—would fascinate us with their memories of olden times. As good fortune would have it (or, was it the artistry of Roborto Dvornicich, Professor of Mathematics at the University of Pisa, who arranged housing for the conference participants) my friends and I stayed at the Villa Kinzica—across the street from the Leaning Tower of Pisa. Over a plate of spaghetti, we could see that tower, one of the "seven wonders of the world" whose very construction took 99 years. And—it REALLY leans! We were charmed by the seven bells, all chiming in different tones. But—most of all—we pictured Galileo Galilei excitedly experimenting with falling bodies . I would be amiss if I did not mention the Botanical Garden of Pisa—situated adjacent to our conference room at POLO DIDATTICO BELLA FACOLTA DI SCIENZE. In the summer of 1543 (the University of Pisa itself was founded in the 12th century) this garden was opened as the first botanical garden in Western Europe. Its present location was taken up 50 years later. While we may not have been able to recognize "METASEQUOIA GLYPTOSTROBOIDES" the peace and serenity of this beautiful park struck chords in all of us. On the third day, the Conference terminated at noon, and we took the bus to Volterra. The bus ride itself ushered in a trip long to be remembered. The incredibly luscious fields of sunflowers and sunflowers—an (Continued on page 331) 289 AN ITERATED QUADRATIC EXTENSION OF GF(2) DOUG WIEDEMANN University of Waterloo, Waterloo, Ontario, Canada (Submitted November 1986) 1 . A CONSTRUCTION It is well known (see, for example, Ex. 3.96 of [1]) that the polynomials x2'3° + x3J + 1 are irreducible in GF(2)[x] for J = 0, 1, 2, . Since (x2'3' + x3J + l)(x3J + 1) = x3J+±+ 1 is a square-free polynomial, it follows that the period of each root of a;2*3 + x + 1 is precisely 3J 1 , only one and a half times the degree of the polyno- mial . The field Cj ~ GF(2)[x]/(x2'3J' + x3' + 1) ~ GF(22'3J) may be obtained by iterated cubic extensions beginning with CQ ~ GF(2)(xQ)9 where x0 £ 1 is a cube root of unity.
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