The Pascal Fibonacci numbers

Transcription

1 Notes o Number Theory ad Discrete Mathematics Vol. 19, 013, No. 3, 5 11 The Pascal Fiboacci umbers J. V. Leyedekkers 1 ad A. G. Shao 1 Faculty of Sciece, The Uiversity of Sydey NSW 00, Australia Faculty of Egieerig & IT, Uiversity of Techology Sydey NSW 007, Australia s: tshao38@gmail.com, Athoy.Shao@uts.edu.au Abstract. The Pascal Fiboacci (PF) umbers for a give Fiboacci umber sum to give the values of that Fiboacci umber. Idividual PF umbers are members of oe of the triagular, tetrahedral or petagoal series or have factors i commo with the pyramidal or other geometric series. For composite umbers, partial sums of PF umbers ca yield a factor, while prime Fiboacci umbers are detected with sums of squares. Keywords: Fiboacci umbers, Biet equatio, Pascal Triagle, Triagular umbers, Tetrahedral umbers, Petagoal umbers, Pyramidal umbers. AMS Classificatio: 11B39, 11B50. 1 Itroductio It is well-kow that the umbers alog the leadig diagoals i the Pascal Triagle sum to umbers i the Fiboacci sequece {F } (with geeralizatios to higher order recursive sequeces ad geometric dimesios [7]). This provides a simple way to calculate the Fiboacci umbers without irratioals, as i the Biet equatio [5, ]; that is by 1 ( p 1) p i F = +. (1.1) i= i 1 We have called elemets of these Pascal Fiboacci (PF) umbers, each of which is give p i N p i = (1.) i 1 which are listed i Table 1 for p from 7 to 59 []. For example, whe p = 17 ad i = 4, the third umber i the sum is N P17 (3) = 8. Similarly, whe p = 43 ad i = 5, N P43 (4) = Agai, whe p = 59, the last i = ½(p 1) =9, so the 8 th umber i the sum is N P59 (8) = = 30! / 8!! =

2 p F p 7 5, 11 9, 8,35, , 45,84,70, , 91,8,495,4,10, , 10,455,1001,187,94,330,45 3 1, 190,99,300,188,8008,435,3003,715, 9 7, 35,300,10,3349,7413,1180,15970,9378,43758,137,180, , 378,95,14950,53130,13459,45157,319770,93930,18475,7558,1854,380, , 51,545,3590,19911,593775,150780, ,4885, , , 70415,11440,319770,544,4845, , 703,7770,58905,343, ,47048, ,010075, , , , ,957700,3870,735471,100947,7315, , 780,9139,73815,435897,194779,7450,181504, ,45140, ,84935,783915,401100, , , ,13459, 8855, , 94,1341,111930,749398, , , ,144030,541885, , , , , ,145475, , ,0075, 3030,150, , 15,1844,194580, ,93819,453790,17737, , , , , , , , , , , ,579840,443515,58595,47500,0475, , 1540,35,3151,8985,035850, , , , , , , , , , , , , , , , , , , ,9019,3145,435 Table 1. Pascal Fiboacci umbers I this paper, we examie the details of the structure of these umbers. I particular, the PF umbers are formed by sequetial ratios of factorials ad therefore have a regular structure. They are composed of primes, the maximum of which is P i 1 for F Pi. For istace, the PF umbers associated with F 19 have the primes {, 3, 5, 7, 11, 13, 17} i varyig proportios. The simple structure makes the positio of each alog the diagoals sigificat. For example, all the first umbers equal (p ), the secod are triagular as are the last, while the third umbers are tetrahedral. The secod last umbers, N i, (i=½(p 5)), are petagoal [, 3]. The secod PF umbers The triagular umbers ca be represeted by T = ½( + 1). (.1) The secod PF umber of each F p (7 p 59) are give by Equatio (.1) with = p 4 (Table ).

3 1 p = p 4 N = ( 1) Table. Secod PF Numbers 3 The third PF umbers The tetrahedral umbers i this cotext [, 3] are give by ad the third PF umbers fit this series with = (p ) (Table 3). H = 1 ( + 1)( + ) (3.1) 1 p = p N = ( + 1)( ) Table 3. Third PF Numbers

4 4 The fourth PF umbers Assumig the patter for cotiues, that is (p i) for, are the fourth umbers compatible with some geometric umber series? Usig = p 8 for the pyramidal umbers [, 3], that is Q = 1 ( + 1)( + 1) (4.1) it is foud that these umbers always have a factor i commo with the fourth PF umbers. I fact, the factor 5 is commo to all fourth PF umbers except F 19, F 9, F 59 ; that is, whe p has a right-ed-digit (RED) of 9 ad is therefore a elemet of the Class 45 Z5 [1, 4, ]. 5 Last three umbers (a) The last umbers are triagular umbers which satisfy Equatio (.1) with = ½(p 1). All these umbers are divisible by 3 so that they fall ito a special subset of the triagular umbers [, 3] (Table 4). p = ½(p 1) N i (Eq.1) ⅓N i Table 4. i = ½(p 3) (b) The secod last umbers N i (i = ½(p 5)) always have 5 as a factor. These umbers are always divisible by 5, ad at least oe of the factors is triagular with varyig from 1 to. Moreover, these umbers are petagoal, give by [, 3]: with = p /4 (Table 5). D = 1 (3 1) (5.1) 8

5 p = p /4 * D i = ½(3 1) positio, i Table 5. Petagoal umbers D i = ½(3 1) ad positio i; *residual of 0.04 for all p is eglected (c) Third last umbers, N i, i = ½(p 7), are always eve (except for p = 3 or 41), ad all are divisible by 7 (11 umbers from p = 17 to 59). Eight of the umbers also have 8 as a factor. Other umbers, N i, with i = 4 to 14 The remaiig umbers have factors i commo with triagular, tetrahedral ad other geometric series, with = p i. They geerally have a factor commo to all, but there are a few exceptios: examples (Table ): i Commo factor for i p Exceptios Factor of N i 4 5 9, ,19,31, ,41, , , , Table. Remaiig umbers, N i 9

6 7 Idetifyig prime F p We have previously [] show how the structure of the Fiboacci umbers ca help to idetify primes. Some alterative complemetary methods are outlied i (a) ad (b) below. 7.1 From PF umbers The PF umbers ca be partially summed to fid a suitable factor (Table 7). p F p Numbers summed Factors S = N + N + N = = S = N + N = = Sum of squares Aother method is via the sum of squares [1]: 1 Table 7. Partial sums of PF umbers F p = d + e. (7.1) Primes oly have oe set of (d, e) with o commo factors. Geerally composites have the same umber of sets as their factors. Oe (d, e) set is give by ( F ) ( F ) 1 F (7.1) p = 1 + ( p + 1) ( p 1) with others from various techiques [1]. Examples are displayed i Table 8. p F p factors d e d, e as F F 4 F F 5 F F 7 F F 8 F , F 10 F F 11 F F 14 F , F 1 F ,149, F 19 F Table 8. Sums of squares Fially, the iterested reader may like to develop geeralizatios of the Pascal Fiboacci umbers which ca be made with suitable geeralizatios of Pascal s triagle [8]. 10

7 Refereces [1] Leyedekkers, J. V., A. G. Shao. Fiboacci Numbers withi Modular Rigs. Notes o Number Theory ad Discrete Mathematics. Vol. 4, 1998, No. 4, [] Leyedekkers, J. V., A. G. Shao. The Structure of Geometric Number Sequeces. Notes o Number Theory ad Discrete Mathematics. Vol. 17, 011, No. 3, [3] Leyedekkers, J. V., A. G. Shao. Geometric ad Pellia Sequeces. Advaced Studies i Cotemporary Mathematics. Vol., 01, No. 4, [4] Leyedekkers, J. V., A. G. Shao. The Structure of the Fiboacci Numbers i the Modular Rig Z 5. Notes o Number Theory ad Discrete Mathematics. Vol. 19, 013, No. 1, 7. [5] Leyedekkers, J. V., A. G. Shao. O the Golde Ratio. Advaced Studies i Cotemporary Mathematics. Vol. 3, 013, No. 1, [] Leyedekkers, J. V., A. G. Shao. Fiboacci ad Lucas Primes. Notes o Number Theory ad Discrete Mathematics. Vol. 19, 013, No., [7] Shao, A. G. Triboacci Numbers ad Pascal s Pyramid. The Fiboacci Quarterly. Vol. 15, 1977, No. 3, 8, 75. [8] Wog, C. K., T. W. Maddocks. A Geeralized Pascal s Triagle. The Fiboacci Quarterly. Vol. 13, 1975, No.,