Notes o Number Theory ad Discrete Mathematics Vol. 19, 013, No. 1, 66 7 The structure of the iboacci umbers i the modular rig Z J. V. Leyedekkers 1 ad A. G. Shao 1 aculty of Sciece, The Uiversity of Sydey Sydey, NSW 006, Australia aculty of Egieerig & IT, Uiversity of Techology Sydey, NSW 007, Australia e-mails: tshao38@gmail.com, athoy.shao@uts.edu.au Abstract: Various iboacci umber idetities are aalyzed i terms of their uderlyig iteger structure i the modular rig Z. Keywords: iboacci sequece, Golde Ratio, modular rigs, Biet formula. AMS Classificatio: 11B39, 11B0. 1 Itroductio Over eight ceturies ago i Chapter XII of his book, Liber Abaci, Leoardo of Pisa (ickamed iboacci) preseted ad solved his famous problem o the reproductio of rabbits i terms of the famous sequece which bears his ame. our ceturies later, Albert Girard i 1634 gave the otatio for the recurrece relatio for the terms of the sequece i use today, amely (1.1) + 1 = +. Over the ceturies sice the iboacci sequece of itegers has bee applied to a myriad of mathematical applicatios, especially i umber theory [1]. I particular, Kepler [6] observed that the ratio of cosecutive iboacci umbers coverges to the Golde Ratio φ. He also showed that the square of ay term differs by uity from the product of the two adjacet terms i the sequece (Simso s or Cassii s Idetity (3.) below). I this paper we aalyse the structure of the iboacci sequece i the cotext of the modular rig Z (Table 1) []. The uderlyig structure accouts for may of the uique properties of this fasciatig sequece, particularly their cogruece properties [7]. 66
Class Row r 0 r 1 +1 r + r 3 +3 r 4 +4 0 0 1 3 4 1 6 7 8 9 10 11 1 13 14 3 16 17 18 19 4 0 1 3 4 Table 1. Rows of modular rig Z Class patters of iboacci umbers The patter of the iboacci umbers i Z is displayed i Table. 1 3 4 6 7 8 9 Z 10 11 1 13 14 16 17 18 19 0 1 3 4 6 7 8 Z 9 30 31 3 33 34 36 37 38 Table. iboacci umbers i Z The patters of the modular residues follow the form N N N M i which the umbers N have the patter ad the iterstitial umbers M have the patter. These patters allow predictio of the class of, ad hece the right-ed-digit (RED) from (Table 3). (1,6) (,7) (3,8) (4,9) (0,) for N for M 19, 39, 9, 79,... 8, 8, 48, 68,... 1, 1, 41, 61,...,, 4, 6,... 14, 34, 4, 74,... 3, 3, 43, 63,... 16, 36, 6, 76,... 17, 37, 7, 77,... 4, 4, 44, 64,... 13, 33, 3, 73,... 6, 6, 46, 66,... 7, 7, 47, 67,... 9, 9, 49, 69,... 18, 38, 8, 78,... 11, 31, 1, 71,... 1, 3,, 7,... 0,,10,,0,... Table 3. Details of the patters ( : Class of ) 67
There are may characteristics of the iboacci sequece that are directly related to this structure. We cosider some of them here. They serve as examples for further aalysis. Aother approach would be to cosider the algebra of ( ) where (x) is the characteristic polyomial associated with the recurrece relatio (1.1) ad is irreducible i the field of its coefficiets [8]. 3 The relatioship to the Golde Ratio If the measure of a lie AB is give by +1,d AB is divided ito two differet sized segmets, AC ad CB, with CB>AC, the AB/CB = CB/AC approximately defies φ, the Golde Ratio if CB = ad AC = -1, so that approximately + (3.1) 1 However, as first oted by Kepler the two sides of (3.1) always differ by uity as we ca see from the class structures (Table 4). ( ) +1 ( ) + 1 4 3,8 9,4 0,,7 0, 41 1,6 1,6 1,6 0, 0, 77,7 4,9 4,9,7 3,8 9 4,9 6,1 3,8 4,9 7, Table 4. Data from Tables, 3 The equality is expressed i Simso s Idetity + 1 = + ( (3.) ) ( ) + 1 / = / + ( ) / (3.3) of which the secod term o the right had side is very small for large ; this is the error term i the iboacci approximatio for φ []. so that which whe substituted ito (3.) yield + + 6 = 4 + 3 (3.4) 4 / 1, (3.) + 6 / + 3 = 3 + + 6 4 + 3, eve, + 1 = (3.6) 3 + 6 + 4 + 3, odd. May other elegat relatioships ca be formed i this way. 68
4 Squarig rectagles A odd umber of golde rectagles with sides equal to successive iboacci umbers ca appear to fit ito squares as demostrated i igure 1. A 8 E B A 8 E 3 3 C C 8 D D 3 B igure 1. Squarig the Golde Rectagle This is ot draw to scale, but essetially a golde rectagle of sides ad 7 uits, ad hece of area 6 square uits, is trasformed ito a square of side 6 uits ad hece of area 64 square uits. Of course, while the eye might just be deceived, Simso is ot! rom Simso s idetity we get for odd that (4.1) 1 + + + 1 = + 1 The structure of the iboacci sequece i Z i Sectio shows that this square sum depeds o the costraits of the squares which occur oly i Classes, ad, ad the sums are also cofied to these classes i harmoy with this square (Table ). Number of Products +1 Classes of +1 3 7 9 69 Class of 1 + + + 1 1 + + = 4 + + = 4 + + = 1 + + = 11 + + = 13 + + = + + = 17 + + = 19 + + = 1 + + = Table. Classes of Sums from (4.1)
The actor 11 The result lim 6 / = 8ϕ + + from [4] was obtaied from the above characteristics of the sequece. Moreover, the Class of the sum of te cosecutive itegers is the same as the class of the seveth umber i the te. The seveth umber times 11 equals the sum of the te. This is cosistet with 11 ad a = (Table 6). Note that the RED of the sum is the same as the RED of the seveth 1 a umber, ad sice the RED of 11 is 1 it is the oly iteger to satisfy. Rage of Class of sum Class of 7 th Iteger, N 7 Class of N 1 10 11 3 1 4 13 14 6 7 16 8 17 9 18 10 19 Table 6. Class structure i sets of 10 itegers The class structure of the 7 th umber i each set of te itegers is: which correspods to 7 o the class patter i Sectio. 7 6 Periodicity of iboacci umber right ed digits A RED periodicity of 60 for itegers was discovered i geeral i 1774 by Joseph Louis Lagrage [6]. However, this periodicity patter is more complicated tha previously assumed for the iboacci sequece. or eve REDs the iterval is 60, but for odd REDs the itervals ca be 0 or 40 which ideed sum to 60, ad for Class the itervals are 30 (Table 7). 70
Class of Δ 0 0 30, 60, 90, 10, 0 30, 30, 30, 30,, 7, 1, 1 30, 30, 30, 30 6 1 1, 81, 141, 01 60, 60, 60, 60 4, 10, 16, 60, 60, 60, 60 8 48, 108, 168, 8 60, 60, 60, 60 9 39, 99, 9, 19 60, 60, 60, 60 1 1 1, 41, 61, 101, 11 40, 0, 40, 0,, 6, 8, 1 0, 40, 0, 40 8 8, 8, 68, 88, 18 0, 40, 0, 40 9 19, 9, 79, 119, 139 40, 0, 40, 0 3 3, 63, 13, 183 60, 60, 60, 60 4 4, 114, 174, 34 60, 60, 60, 60 6 36, 96, 6, 16 60, 60, 60, 60 7 7, 117, 177, 37 60, 60, 60, 60 7 3 3, 43, 83, 103, 143 0, 40, 0, 40 4 14, 34, 74, 94, 134 0, 40, 0, 40 6 16, 6, 76, 116, 136 40, 0, 40, 0 7 17, 37, 77, 97, 137 0, 40, 0, 40 8 3 33, 93, 3, 13 60, 60, 60, 60 4 4, 84, 144, 04 60, 60, 60, 60 6 6, 66, 16, 186 60, 60, 60, 60 7 7, 87.147, 07 60, 60, 60, 60 3 3 13, 3, 73, 113, 133 40, 0, 40, 0 4 4, 44, 64, 104, 14 40, 0, 40, 0 6 6, 46, 86, 106, 146 0, 40, 0, 40 7 7, 47, 67, 107, 17 40, 0, 40, 0 4 1 1, 111, 171, 31 60, 60, 60, 60 1, 7, 13, 19 60, 60, 60, 60 8 18, 78, 138, 198 60, 60, 60, 60 9 9, 69, 19, 189 60, 60, 60, 60 9 1 11, 31, 71, 91, 131 0, 40, 0, 40 3,, 9, 11, 0, 40, 0, 40 8 38, 8, 98, 118, 8 0, 40, 0, 40 9 9, 49, 89, 109, 149 0, 40, 0, 40 Table 7. Periodicities of iboacci REDs 71
7 ial commets The structure of the sequece is 1 4 4 0 4 4 0 4 4 0 4 4 which follows from the restricted distributio of the squares i Z. This simple structure facilitates the formatio of Pythagorea triples from [1], ad kow results such as ad, 1 = + >, + = + j= 1 1 j. These ca also be related to the structure. ially, the iterested reader might like to apply the foregoig to the Pellia sequeces to compare the similarities ad differeces with the iboacci sequece [3]. Refereces [1] Leyedekkers, J.V., A.G. Shao, J.M. Rybak. 007. Patter Recogitio: Modular Rigs ad Iteger Structure. North Sydey: Raffles KvB Moograph No 9. [] Leyedekkers, J.V., A.G. Shao. The Modular Rig Z. Notes o Number Theory ad Discrete Mathematics. Vol. 18, 01, No., 8 33. [3] Leyedekkers, J.V., A.G. Shao. Geometrical ad Pellia Sequeces. Advaced Studies i Cotemporary Mathematics. Vol., 01, No. 4, 07 08. [4] Leyedekkers, J.V., A.G. Shao. O the Golde Ratio (Submitted). [] Leyedekkers, J.V., A.G. Shao. The Decimal Strig of the Golde Ratio (Submitted). [6] Livio, Mario. The Golde Ratio. Golde Books, New York, 00. [7] Shao, A.G., A.. Horadam, S.N. Colligs. Some iboacci Cogrueces. The iboacci Quarterly. Vol. 1, 1974, No. 4, 1 4. [8] Simos, C.S., M. Wright. iboacci Imposters. Iteratioal Joural of Mathematical Educatio i Sciece ad Techology. Vol. 38, 007, No., 677 68. [9] Ward, M. The Algebra of Recurrig Series. Aals of Mathematics. Vol. 3, 1931, No. 1, 1 9. 7