
THE OFFICIAL JOURNAL OF THE FIBONACCI"ASSOCIATION VOLUME 24 PAY NUMBER 2 1986 CONTENTS Acknowledgements 98 On Some Polygonal Numbers Which Are, at the Same Time, the Sums, Differences, and Products of Two Other Polygonal Numbers Shoichi Hirose 99 Letter to the Editor O. William McClung 106 Some Results Concerning Pythagorean Triplets R. M. Sternheirner 107 On the Minimum of a Ternary Cubic Form Werner Georg Nowak 129 On the Coefficients of a Recursion Relation for the Fibonacci Partition Function , . Tad White 133 Higher-Order Fibonacci Sequences Modulo M DerekK. Chang 138 A Note on Pascal-TTriangles, Multinomial Coefficients, and Pascal Pyramids Richard C. Bollinger 140 Research Conference Announcement 144 An Entire Function That Gives the Fibonacci Numbers at the Integers A. W. Goodman 145 On the Enumerator for Sums of Three Squares John A. Ewell 150 Bernoulli Numbers and Kummer's Criterion Harlan R. Stevens 154 Discovering Fibonacci Identities Calvin T. Long 160 Fibonacci Numbers and Their Applications 167 Explicit Formulas for Numbers of Ramanujan F. T. Howard 168 Skew Circulants and the Theory of Numbers: An Addendum L J. Good 176 On Fibonacci Binary Sequences Derek K. Chang 178 Elementary Problems and Solutions. Edited by A. P. Hillman, Gloria C. Padilla, & Charles R. Wall 180 Advanced Problems and Solutions Edited by Raymond E. Whitney 185 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 MATHEMATICS,, SOUTH DAKOTA STATE UNIVERSITY, BOX 2220, BROOKINGS, SD 57007-1297. 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 $25 for Regular Membership, $35 for Sustain Mem- bership, and $65 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. 1986 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. tjfe Fibonacci Quarterly Founded in 1963 by Verner E. Hoggatt, Jr. (1921-1980) Br. Alfred Brousseau, and I.D. Ruggles THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION DEVOTED TO THE STUDY OF INTEGERS WITH SPECIAL PROPERTIES EDITOR GERALD E. BERGUM, South Dakota State University, Brookings, SD 57007 ASSISTANT EDITORS MAXEY BROOKE, Sweeny, TX 77480 PAUL F. BYRD, San Jose State University, San Jose, CA 95192 LEONARD CARLITZ, Duke University, Durham, NC 27706 HENRY W. GOULD, West Virginia University, Morgantown, WV 26506 RODNEY HANSEN, Whitworth College, Spokane, WA 99251 A.P. HILLMAN, University of New Mexico, Albuquerque, NM 87131 A.F. HORADAM, University of New England, Armidale, N.S. W. 2351, Australia DAVID A. KLARNER, University of Nebraska, Lincoln, NE 68588 RICHARD MOLLIN, University of Calgary, Calgary T2N 1N4, Alberta, Canada JOHN RABUNG, Randolph-Macon College, Ashland, VA 23005 DONALD W. ROBINSON, Brigham Young University, Provo, UT 84602 LAWRENCE SOMER, George Washington University, Washington, DC 20052 M.N.S. SWAMY, Concordia University, Montreal H3C 1M8, Quebec, Canada D.E. THORO, San Jose State University, San Jose, CA 95192 THERESA VAUGHAN, University of North Carolina, Greensboro, NC 27412 CHARLES R. WALL, Trident Technical College, Charleston, SC 29411 WILLIAM WEBB, Washington State University, Pullman, WA 99163 BOARD OF DIRECTORS OF THE FIBONACCI ASSOCIATION CALVIN LONG (President) Washington State University, Pullman, WA 99163 G.L. ALEXANDERSON University of Santa Clara, Santa Clara, CA 95053 HUGH EDGAR (Vice-President) San Jose State University, San Jose, CA 95192 MARJORIE JOHNSON (Secretary-Treasurer) Santa Clara Unified School District, Santa Clara, CA 95051 JEFF LAGARIAS Bell Laboratories, Murray Hill, NJ 07974 ACKNOWLEDGEMENTS In addition to the members of the Board of Directors and our Assistant Editorss the following mathematicians9 engineers5 and physicists have assisted The Fibo- nacci Quarterly by refereeing papers during the past year. Their special efforts are sincerely appredicted. We apologize for any names that have inadvertently been overlooked. George E. Andrews Helaman Ferguson R. M„ Najar Pennsylvania State U. Brigham Young U. U. of Wisconsin/Whitewater Krassimir Atanassov James Flanigan Norman Richert Bulgarian Acad, of Sci. Pacific Palisades9 CA U. of Houston Robert Backstrom Joseph Gallian Ernst Selmer GYMEA, N.S.W. U. of Minnesota/Duluth U. of Bergen Anthony E. Barkauskas Peter Hagis J. 0. Shallit U. of Wisconsin/LaCrosse Temple U. U. of Chicago Bruce C. Berndt P. Te Howard Tony Shannon U. of Illinois Wake Forest U* U. of Wales George Berzsenyi Steven Kahan Donald Snow Lamar U. Queens College Brigham Young U. David Bressoud Dan Kalman Lawrence Somer Pennsylvania State U. Rancho Palos Verdes9 CA Washington, D.C. Richard Brualdi J. R. Kerr Peter St. John U. of Wisconsin National U. of Singapore Rutgers U. John Burke Clark Kimberling M. V. Subbarao Gonzaga U. U. of Evansville U. of Alberta Dario Castellanos Clayton Knoshaug Richard Taylor Valencia, Venezuela Bemidji State U. Princeton U. M. J. DeLeon Gunter Kohler Dmitri Thoro Florida Atlantic U. U. of Wurzburg San Jose State U. Naisingh Deo Joseph Konhauser Stephen Turner Washington State U. Macalester College Babson College Underwood Dudley Steve Ligh Marcellus Waddill DePauw Un. U. of Southwestern LA Wake Forest U. Ernest Eckert Pin-Yen Lin Albert Wilansky Aalborg U. Center Republic of China Lehigh U. E. A. Enneking W. L. McDaniel Kenneth Yocom Portland State U. U. of Missouri/St. Louis South Dakota State U. John Ewe11 Richard Mollin Hisashi Yokota Northern Illinois U. U. of Calgary South Dakota State U. 98 [May ON SOME POLYGONAL NUMBERS WHICH ARE^ AT THE SAME TIME^ THE SUMS^ DIFFERENCES.* AND PRODUCTS OF TWO OTHER POLYGONAL NUMBERS SH01CHI HIROSE Mita High School, Tokyo 108 Japan (Submitted December 1981) We denote the nth ^-gonal number by Pn,g = n^(9 ~ 2)n - (g - 4)}/2. f For g = 39 5S 6, and 8, we denote Pn,g by Tns the triangular numbers, P n9 the pentagonal numbers, Hn9 the hexagonal numbers, and 0n, the octagonal numbers, respectivelyo We denote Pntg by Pn whenever there is no danger of confusion* Sierpinski [18] has proved that "there exist an infinite number of trian- gular numbers which are, at the same time, the sums, differences and products of two other triangular numbers> 1.tf Ando [1] proved that "there exist an in- finite number of ^-gonal numbers that can be expressed as the sum and differ- ence of two other ^-gonal numbers at the same time." It was also shown in [6] that there are an infinite number of ^-gonal numbers that can be expressed as the product of two other ^-gonal numbers. The present paper will show that there are infinitely many ^-gonal numbers (#=5,6, and 8) which are at the same time the sums, differences, and products of two other #-gonal numbers. 1. THE EQUATION Pu + W + Pv + W = Pu+v + w If Px + Py = Pz 9 by putting u = z - y 9 v = z - x 5 and w - x + y - z9 we have x = u + w9 y = V + W9 and z = u + V + w. However, a little algebra shows that P P P u + w + v + w = u+v + w implies 2(g - 2)uv = (g - 2)w(w - 1) + 2w. Hence Theorem 1: Any solution x9 y9 z of the equation Px + Py = Pz can be expressed as x ~ u + w9 y = v + w9 z = u + v + w9 where w = 0 (mod g - 2) and uv = {(g - 2)w2 - (g - *)w}/2(g - 2). Using this theorem, which is a generalization of the work of Fauquembergue [7] and of Shah [15] on triangular numbers, we can obtain the solutions of the equation Px + Py - Pz in an efficient way.
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