^h ^Fibonacci Quarterly THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION VOLUME 15 [l^WST NUMBER 1 CONTENTS Residues of Generalized Fibonacci Sequences C. C. Yalavigi 1 Composites and Primes Among Powers of Fibonacci V. E. Hoggatt, Jr., and Numbers, Increased or Decreased by One Marjorie Bicknell-Johnson 2 Divisibility by Fibonacci and Lucas Squares V. E. Hoggatt, Jr., and Marjorie Bicknell-Johnson 3 Letter to the Editor . John W. Jameson 8 An Elementary Proof of Kronecker's Theorem J. Spencer 9 Fibonacci Numbers in the Count of Spanning Trees Peter J. Slater 11 The Diophantine Equation 3 3 3 (x1 + x2 + - +xnf = x + x 2 + - + x . W. R. Utz 14 An Application of W. Schmidt's Theorem Transcendental Numbers and Golden Number Maurice Mignotte 15 The Reciprocal Law W. E. Greig 17 Binet's Formula Generalized A. K. Whitford 21 On the Multinomial Theorem David Lee Hilliker 22 A Fibonacci Formula of Lucas and its Subsequent Manifestations and Rediscoveries . H. W. Gould 25 Numerator Polynomial Coefficient Arrays for Catalan V. E. Hoggatt, Jr., and and Related Sequence Convolution Triangles. Marjorie Bicknell-Johnson 30 Fibonacci-Like Groups and Periods of Fibonacci-Like Sequences .Lawrence Somer 35 Solution of a Certain Recurrence Relation .Douglas A. Fults 41 On Tribonacci Numbers and Related Functions Krishnaswami Alladi and V. E. Hoggatt, Jr. 42 Sums of Fibonacci Reciprocals W. E. Greig 46 Fibonacci Notes — 5. Zero-One Sequences Again L. Carlitz 49 On the A/"-Canonical Fibonacci Representations of Order N . Robert Silber 57 A Rearrangement of Series Based on a Partition of the Natural Numbers H. W. Gould 67 n n k A Formula for £ ' Fk (x)y " and its Generalization. M. N. S. Swamy 73 7 Some Sums Containing the Greatest Integer Function L Carlitz 18 Wythoff's Nim and Fibonacci Representations Robert Silber 85 Advanced Problems and Solutions Edited by Raymond E. Whitney 89 Elementary Problems and Solutions Edited by A. P. Hillman 93 FEBRUARY 1977 tfie Fibonacci Quarterly THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION DEVOTED TO THE STUDY OF INTEGERS WITH SPECIAL PROPERTIES EDITOR V. E. Hoggatt, Jr. EDITORIAL BOARD H. L. Alder Gerald E. Bergum David A. Klarner Marjorie Bicknell-Johnson Leonard Klosinski Paul F. Byrd Donald E. Knuth L. Carlitz C. T. Long H. W. Gould M. N. S. Swamy A. P. Hillman D. E. Thoro WITH THE COOPERATION OF Maxey Brooke L. H. Lange Rro. A. Brousseau James Maxwell Calvin D. Crabill Sister M. DeSales T. A. Davis McNabb Franklyn Fuller D. W. Robinson A. F. Horadam Lloyd Walker Dov Jarden Charles H. Wall The California Mathematics Council All subscription correspondence should be addressed to Professor Leonard Klosinski, Mathematics Department, University of Santa Clara, Santa Clara, California 95053. All checks ($15.00 per year) should be made out to the Fibonacci Association or The Fibonacci Quarterly. Two copies of manuscripts intended for publication in the Quarterly should be sent to Verner E. Hoggatt, Jr., Mathematics Department, San Jose State University, San Jose, California 95192. All manuscripts should be typed, double-spaced. Drawings should be made the same size as they will appear in the Quarterly, and should be done in India ink on either vellum or bond paper. Authors should keep a copy of the manuscript sent to the editors. The Quarterly is entered as third-class mail at the University of Santa Clara Post Office, California, as an official publication of the Fibonacci Association. The Quarterly is published in February, April, October, and December, each year. Typeset by HIGHLANDS COMPOSITION SERVICE P. 0 . Box 760 Clearlake Highlands, Calif. 95422 RESIDUES OF GENERALIZED FIBONACCI SEQUENCES C. C. YALAVIGI Government College, Mercara, India Consider a sequence of GF numbers, [wn(b,c; P.Q)}^. For b =c= 1, L Taylor [1] has proved the follow- ing theorem. Theorem. The only sequences which possess the property that upon division by a (non-zero) member of that sequence, the members of the sequence leave least +ve, or -ve residues which are either zero or equal in absolute value to a member of the original sequence are the Fibonacci and Lucas sequences. Our objective is to consider the extension of this theorem to GF sequences by a different approach, and show that a class of sequences can be constructed to satisfy the property of this theorem in a restricted sense, i.e., for a particular member only. For convenience, wn(b,l; 0,1), wn(b,1; 2,b), wn(b,1; P,Q) shall be designated by u v n> n, Hnr respectively. r Let Hk+r = (-l) ~ Hk.r (mod H k). Assume without loss of generality, k to be +ve. We distinguish 2 cases: (A)0<r<k, and (B) r>k. (A) Evidently, the members leave least residues which are either zero or equal in absolute value to a member of the original sequence. (B) Allow |//_ 5 + ; | < \Hk\ < \H.S\. Let k+S 1 k+s (D H2k+S - (-V ' H.S (mod Hk), H2k+S+1 = (-V H.s.j (mod Hk). Clearly, the property of above-cited theorem holds for {Hp}^, iff k+s (_1)k+s-lH_s _ HQ ( m o d Hj<)i a n d (-D H-s-i = HQ + 1 (mod Hk), for some c such that -s + 1 < e < 2k. Denote the period of { Hn (mod Hk)} Q by k(Hk). Rewrite the given sequence as { Hp'}™^, where H'n' = Hn. Set k' = k + t, s' = s - t, and 2'=£ + t. Then, it is easy to show that k(Hk) = 2k + s-z, k(H'k>) = 2k'+ s'- c', and k(Hk) = k(H'k-). We assert that ^ / / ^ ' ^ is even, for f = ^ - z)/2 obtains s'= si', k(H'k') = 2k', and the substitution ois-s. = 2t + 1 leads to s' - si' = 1, k(H'k') = 2k'+ 1, which is a contradiction. Hence, it is sufficient to examine the following system of congruences, viz., (2) H'2k- = H'0 (mod H'k-\ H'2k+1 = H', (mod H'k>). These congruences imply k+t 1 k 1 (3) H2k+t = Ht = (-V ~ H^(mo6Hk) = (-l) ~ {Ht- (2Q - bP)ut } (mod Hk ) k 1 = (-1) ~ {Pvt-Ht}(m^Hk). Therefore, (i) P = 0, Q = 1, and (\\) P = 2, Q = b, furnish readily the desired sequences, and they are the only sequences for which the property of L. Taylor's theorem holds. For the restripted case, by using the well known formula Hn = Pun-i + Qun, it is possible to express H.s = HQ (mod Hk), and //_s_; =H%+1 (mod Hk) as two simultaneous equations in P, Q, and obtain their solution for given s, si, and k. In particular, the latter case may be handled by using k(H'k') = k(uk'), where H'k' is selected arbitrarily to satisfy k'= k(uk')/2 and Hk- = Puk-„i + Quk, determines/3 and Q. Example: H'g = 19, k(H'9) =18, P = 9, Q = -5. 1 2 RESIDUES OF GENERALIZED FIBONACCI SEQUENCES FEB. 1977 REFERENCES 1. L Taylor, ''Residues of Fibonacci-Like Sequences," The Fibonacci Quarterly, Vol. 5, No. 3 (Oct. 1967), pp. 298-304. 2. C. C. Yalavigi, "On a Theorem of L Taylor,"Math. Edn., 4 (1970), p. 105. ******* COMPOSITES AND PRIMES AMONG POWERS OF FIBONACCI NUMBERS, INCREASED OR DECREASED BY ONE V. E. HOGGATT, JR., and MARJORIE BICKNELL\IOHNSON San Jose State University, San Jose, California 95192 It is well known that, among the Fibonacci numbers Fn, given by F-i = 1 = F2, Fn+1 * Fn + Fn-i, 2 Fn + 1 is composite for each n > 4, while Fn - /is composite for/7 > 7. It is easily shown that Fn ± 1 is also composite for any n, since Fn± 1 = Fn-2Fn+2, Fn + / = Fn+1Fn^1 . Here, we raise the question of when F™ ± 1 is composite. First, if k£0 (mod 3), then Fk is odd, F™ is odd, and F™± 7 is even and hence composite. Now, suppose we n n m deal with F™k ± I Since A - B always has (A - B) as a factor, we see that F™k - 1 is composite except when (A- B)= 1; that is, for k= 1. Thus, Theorem 1. F™ - 1 is composite, k ± 3. m m m We return to F™k + I For/77 odd, then A + B is known to have the factor (A + B), so that F™k + 1 has the factor (F^k + 11 and hence is composite. If m is even, every even m except powers of 2 can be written in the form (2j + 1)2' = m, so that + m 2 2i+1 2 2i+1 F3k 1 = (F 3k) + (1 ) . m m which, from the known factors of A + B , m odd, must have (Fjk + 1) as a factor, and hence, F™k + /is composite. 2 This leaves only the case F™k + /, where m = 2'. When k= 1, we have the Fermat primes,? + /, prime for / = 0, 1, 2, 3, 4 but composite for / = 5, 6. It is an unsolved problem whether or not 22' + 1 has other prime 3 m 3 m 3 values. We note in passing that, when k = 2,F6=8 = 2 , and 8 ± 1 = (2 ) ^ ± 1 = (2 ) ± 7 is always com- 3 posite, since A ± B is always factorable. It is th ought that Fg + 1 is a prime. Since F3k=Q (mod 10),Ar=0(mod 5 ) , / ^ , + / = 102'-t+1. Since F?/, = 6 (mod 10), i> 2, k^ 0 (mod 5), F2' + 1 has the form 10t + 7, k4ft (mod 5). We can sum- marize these remarks as Theorem 2.