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International Journal of Engineering & Technology, 7 (4.10) (2018) 1012-1014 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET Research paper Gaussian Pell Numbers P. Balamurugan1*, A. Gnanam2 1 Department of Science and Humanities, M. Kumarasamy College of Engineering (Autonomous), Karur – 639113. 2 Departments of Mathematics, Government Arts College, Trichy – 620022. *Corresponding author E-mail: [email protected] Abstract Gaussian numbers means representation as Complex numbers. In this work, Gaussian Pell numbers are defined from recurrence relation of Pell numbers. Here the recurrence relation on Gaussian Pell number is represented in two dimensional approach. This provides an extension of Pell numbers into the complex plane. Keywords: Pell sequence, Gaussian integers, Recurrence relations, Gaussian Pell number. The first few terms of the Pell sequence are 0,1,2,5,…….Here, in this 1. Introduction paper the Gaussian Pell sequence is defined by a recurrence relation. “God invented the integers; all else the work of man” as quoted 2. The gaussian pell number by Kronecker the research on spotting subsets of integers that follow recurrence relations is a delightful and inexhaustive one. It The set of Gaussian Pell numbers is denoted by GP n, m and is is actually a wide class of problem. Rather than as a science defined in analogy with the Pell recurrence relation Mathematics is thought as a creative art by Mathematics. The theory of numbers has always occupied a unique position in PPPP2 , 0 and P 1 (1) the world of mathematics. Noted mathematicians on one hand n2 n 1 n 0 1 and numerous amateurs on the other hand share a similar interest and are attracted not towards any other theory but towards the Then the two-dimensional recurrence relations, satisfied by theory of numbers. GP n, m will be Number theory, regarded ‘Queen of Mathematics’ provides all the weapons of mathematics and so both professional mathematician and upcoming research scholars are invariably GPn 2, m 2 GPn 1, m GPnm , (2) attracted to number theory. Also, it is a fact that many important branches of mathematics has their origin in number theory. As the GPnm , 2 2 GPnm , 1 GPnm , (3) mathematician Sierpinski once said the progress of knowledge of numbers is advanced not only by what is known already but the where realization of what yet to be known. Leonardo Fibonacci, Mathematical innovator of the 13th Century was a solitary flame of mathematical genius during the middle GP0,0 0, GP 1,0 1, GP 0,1 i , GP 1,1 1 i (4) ages. Fibonacci wrote in 1202, the Liber Abaci, in which he explained the Hindu Arabic Numerals and how they are used in with GP n, m n im Computation. Fibonacci is remembered particularly for the sequence of number 1,1,2,3,5,…. to which his name has been applied. The conditions are sufficient to specify unique value of This sequence is the subject of continuing research especially by GP n, m at each point nm, in the plane . the association which publishes the Fibonacci Quarterly. Recursive when m 0 , th definition of the n Fibonacci number Fn is GP n2,0 2 GP n 1,0 GP n ,0 Fn F n12 F n ,3 n and FF121. Using the Fibonacci recurrence relation and different initial and hence conditions, many integral sequences can be constructed. A.F. Horadam [5] by introduced the concept of Complex Fibonacci GP n,0 Pn (5) numbers. C.J. Harman in [1] extended the complex Fibonacci numbers and established some recurrence relations concerning them. GP n,1 2 PGP 1,1 P GP 0,1 As in Fibonacci sequence and Lucas sequence the sequence of nn1 Pell numbers is also expressed as by the recurrence relation By substitution, PPPPn22 n 1 n , 0 0 and P1 1 GP n,1 2 Pnn 1 i iP 1 Copyright © 2018 Authors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1013 International Journal of Engineering & Technology 2k 22pn i p n p n1 (6) 2 1i PPn j m j GPn 2 km , 2 k GPnm , j1 and so by (1) and so by (7) 2k GPnm , PGPnmm ,1 P1 GPn ,0 1 PPnjmj 1 iPP nkmk 2 2 1 PP nm 1 iP nkmk 2 1 P 2 iPP nm 1 j1 4 Pm2 P n i 2 P n P n11 P m GP n ,0 Equating real and imaginary parts, P2 P iP P P m n n11 m n 2P P iP P P P m n m n11 m n PPPPPPPPnkmk2 2 1 nkmk 2 1 2 nm 1 nm 1 0 (13) GP n, m Pn P m11 iP m P n (7) and 2k Here GP n,0 P , and GP0, m ip . 1 n m PPPPPPPPPPnjmj nkmk 2 2 1 nm 1 nkmk 2 1 2 nm 1 (14) j1 4 3. Recurrence equations and identities Substitution of Ppn2 k m 2 k 1 from (13) into (14) gives Combination of (2) and (3) gives 2k 2 PPPPPP (15) GPn 2, m 2 Rn 2 GPn 1, m 2 GPnm , 2 n j m j n 2 k 1 m 2 k n 1 m j1 Im 2 GP n 2, m 1 GP n 2, m (8) Similarly, which is an interesting two dimensional version of Pell 21k 2 PPPPPP (16) recurrence relation and gives the growth-characteristic of the n j m j n 2 k 2 m 2 k 1 n m 1 j1 numbers in a unique fashion: Gaussian Pell number GP n, m is defined as: ''n is the sum of twice the real part of previous Identities (15) and (16) unify and generalize certain identities and vertex and the real part vertex that precedes it along the real axis provides some examples as special cases. and ''m is the sum of twice the imaginary part of the previous For example, nm0 yields the well known identity: vertex and imaginary part of the vertex that precedes it along the 2 2 2 2PPPPP1 2 ... NNN 1 imaginary axis on the Gaussian lattice. From (15), in this case mn0, 1 gives GP n1, m 1 Pn1 P m 2 iP m 1 P n 2 2PP1 2 P 2 P 3 ... P 2k P 2 k 1 P 2 k P 2 k 2 = Pn122 P m 1 P m i P n 1 P n P m 1 From (16), in this case nm0, 1 gives 21Pn1 P m 1 i P m P n 1 iP n P m 1 2PP P P ... P P P2 By (7), we have 1 2 2 3 2k 1 2 k 2 2 k 2 By the choice of various parameters we can get many identities. For example, equation (15) with mn0, 2 gives GPn 1, m 1 2 PPnm11 1 iGPmn , (9) 2PP1 3 P 2 P 4 ... P 2k P 2 k 2 P 2 k P 2 k 3 Repeatedly applying (9), we get From (16), in this case mn2, 0 gives 2PP13 P 24 P ... P 2123k P k P 2223 k P k GPn 1, m 2 2 PPn1 m 2 1 iPP n 1 m 1 iPP n m 2 Taking mk2 1, in (13) we get GPn 2, m 2 2 1 iPP n2 m 2 PP m 1 n 2 iPP n 1 m 2 2 1 i Pn2 P m 2 P m 1 2 P n 1 P n iP n 1 2 P m 1 P m PPPPPPPPnkmk2 1 2 2 nk 2 2 mk 2 1 nm 1 nm 1 (17) 2 1 iPP n2 m 2 2 PP n 1 m 1 PP n m 1 2 iPP n 1 m 1 iPP n 1 m and together (13) and (17) constitute a generalization of some 2 1 iPP 2 PP 1 iPP iPP n2 m 2 n 1 m 1 n m 1 n 1 m well-known classical identities. For example if nm1, 0 they give GPn 2, m 2 2 1 iPP PP GPnm , (10) N n2 m 2 n 1 m 1 PPPN2 1 , 1. NNN11 Repeatedly applying (9) and (10), we get with nm1, 2 equations (13) and (17) yield the identity N PPPPNNNN1 1 2 2 5 1 . GPn 3, m 3 2 1 iPP n3 m 3 PP n 2 m 2 GPnm 1, 1 GPn 4, m 4 2 1 iPP PP GPn 2, m 2 n4 m 4 n 3 m 3 4. Conclusion 2k GPn 2 km , 2 k 2 1 i PPn j m j GPnm , (11) j1 Fibonacci and Lucas numbers have always been an enthusiastic material to study for both amateur and established mathematicians. 21k Also, Gaussian approach is adding feather to it. Combination of (12) GPnk 2 1, mk 2 1 2 1 i PPn j m j GPmn , these two stimulates the researchers to extend the study to other j1 sequence of numbers which may be given by recurrence relations. So, I have tried the single case of Pellian numbers. Further study From (11), will be made in future for other sequence of numbers. 1014 International Journal of Engineering & Technology References [1] Harman CJ (1981) , Complex Fibonacci Numbers, The Fibonacci Quaterly, 13, 82-86. [2] Berzsenyi G (1975), Sums of Products of Generalized Fibonacci Numbers, The Fibonacci Quaterly, 13, 343-344. [3] Berzsenyi G (1977) , Gaussian Fibonacci Numbers, The Fibonacci Quaterly, 15 , 233-236.
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  • Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and Their Explicit Formulas by Ian Walker Ju

    Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and Their Explicit Formulas by Ian Walker Ju

    Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and their Explicit Formulas By Ian Walker June 2011 MST 501 Research Project In partial fulfillment for the Master’s of Teaching Mathematics (M.S.T.) Portland State University, Portland, Oregon John Pell (1611-1685) 2 John Pell (1611-1685) An “obscure” English Mathematician • Part of the 17th century intellectual history of England and of Continental Europe. • Pell was married with eight children, taught math at the Gymnasium in Amsterdam, and was Oliver Cromwell’s envoy to Switzerland. • Pell was well read in classical and contemporary mathematics. • Pell had correspondence with Descartes, Leibniz, Cavendish, Mersenne, Hartlib, Collins and others. • His main mathematical focus was on mathematical tables: tables of squares, sums of squares, primes and composites, constant differences, logarithms, antilogarithms, trigonometric functions, etc. 3 John Pell (1611-1685) An “obscure” English Mathematician • Many of Pell’s booklets of tables and other works do not list himself as the author. • Did not publish much mathematical work. Is more known for his activities, correspondence and contacts. • Only one of his tables was ever published (1672), which had tables of the first 10,000 square numbers. • His best known published work is, “An Introduction to Algebra”. It explains how to simplify and solve equations. • Pell is credited with the modern day division symbol and the double-angle tangent formula. • Pell is best known, only by name, for the Pell Sequence and the Pell Equation. 4 John Pell (1611-1685) An “obscure” English Mathematician • Division Symbol: • Double-Angle Tangent Formula: 2tan tan2 2 • Pell Sequence: 1 tan p 2p p p 1, p 2,n 2 n n1 n2 0 1 • Pell Equation: 2 2 x 2y 1 5 John Pell (1611-1685) An “obscure” English Mathematician • Both the Pell Sequence and the Pell Equation are erroneously named after him.