International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 4, Issue 11 November 2017

Pisot Numbers Property by Fibonacci sequence

Dr. M Anita Institute of Aeronautical Engineering College Mr. G Nagendra Kumar Institute of Aeronautical Engineering College Ms. P Rajani Institute of Aeronautical Engineering College

ABSTRACT Unit quadratic pisot number is a pisot number for which the product with its conjugates is  1, and they must have the minimal polynomials of degree 2.These numbers will satisfy the polynomials of the form x2 – cx - 1 for c  { 1,2,3….} or x2 – cx + 1 for c  {3,4,5….} Let q be a unit quadratic pisot number lm (q) = | Dq- c | where ‘m’ a particular integer and C/D is the best approximation of q.In this paper, we present the best approximation by Fibonacci sequence by considering golden number is 1.366 KEYWORDS Pisot Numbers: :Fibonacci sequence:

INTRODUCTION Pisot numbers have a long history, being studied as early as 1912. Salem first got interested in Pisot numbers q because of their property that q n 0 (mod 1) as n  . Salem shows that the set of pisot numbers is infinite and are the only algebraic numbers that have this property. lm(q) is given by m n n-1 l (q) = inf {|y|:y = enq + en-1 q +……+ e0,ei {  m,  (m-1)..,  1, 0 } y  0 } [3] Definition:An is a root of a polynomial with integer coefficients.[1] Definition:An is a root of a monic polynomial with integer coefficients.[1] Definition:The Conjugates of an algebraic number are the other roots of the algebraic numbers minimal polynomial.[1] Definition:A pisot number is a positive real algebraic integer greater than 1, all of whose conjugates are of modulus strictly less than 1.[1] Definition: A Salem number is a positive real algebraic integer greater than 1, all of whose conjugates are of modulus less than or equal to 1. And at least one of the conjugates must be of modulus 1.[1]

Definition:The Fibonacci sequence (Fk) defined as follows.

F0 = 0, F1= 1, Fk = Fk-1 + Fk-2, K = 2, 3…..

THEOREM: k-2 k-1, m IfA < m A for some integer K  1,thenl (A) = | FkA - Fk+1| Proof: To prove the theorem, first we prove following two lemmas. k-2 m Lemma: If m > A , then l (A)  | Fk+1 – FkA | k-1 m Lemma: If m  A , then l (A) | Fk+1 – FkA | 1072 Dr. M Anita, Mr. G Nagendra Kumar, Ms. P Rajani International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 4, Issue 11 November 2017

We use the following identities n A = Fn A + Fn-1 for n = 0,1……. a Fa+1Fb -Fa Fb+1 = ( -1) Fb-a if b a  0,

Fa+1 Fb+1 + FaFb = Fa+b+1 if a, b  0. and the problem has a solution if and only if m>Ak-2. The solutions are given by the formulas e Fn k-1 - (F n  1 -1)m e (Fn 2 -2) m- F n  k n , n 1= with m  Fk+1 if n = 1 F m k2 if n = 2 2 F F nk  m  nk if n  3 Fn2 1 Fn2  3 The proof Lemmais given in three cases. m Case I:Supposethat K 3 and assume on the contrary thatl (A) < | Fk+1 – FkA | Case II:The number Z has necessarily the form n n-1 n-2 n-3 n Z = en A + en-1 A + m (A – A +…..+ (-1) )with n  3 and - m  en-1 - en F Case III: If m  k2 , thenlm (A) | F - F A | 2 k+1 k

Now consider the number Yi = Fi+1- Fi A, 0 < | yk+1 | < | yk|< | yk-1 | andykand yk-1 have opposite signs we have yk+1 = yk + yk-1

By the definition of the Fibonacci sequence 0< | Z | < |yk | F By the definition of z, m  k3 2

| Z |  | Fk+2- Fk+1 A |= |yk+1 | k-1 Since m  A and m < Fk+2

|yk+1 |  | Z | and therefore|yk+1 |< | Z | < | yk | n Now we distinguish two cases.If Z and (-1) yk have the same sign, then n+1 Z = Z + (-1) yk n n-1 n-2 n-3 n 2 = en A + en-1 A +m (A - A +….+ (-1) A ) n+1 n + (-1) (m-Fk ) A + (-1) (Fk+1 - m )

Satisfies 0< | Z | < | yk | F further Z belongs to mand k1 F  F 2 k k+1

Finally, since the equalities | m-Fk | = m and | Fk+1 - m | = m cannot hold Simultaneously, Z is smaller than Z. This contradicts the definition of Z. n n If Z and (-1) yk have different signs, then Z and (-1) yk-1 have the same sign. Then it follows that n+1 Z =Z + (-1) yk-1

1073 Dr. M Anita, Mr. G Nagendra Kumar, Ms. P Rajani International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 4, Issue 11 November 2017

n n-1 n-2 n-3 n 2 n+1 n = en A + en-1 A + m (A -A +..+(-1) A )+ (-1) (m-Fk-1) A+(-1) (Fk-m) satisfies 0 < | Z | < | yk-1 - yk+1 | = | yk | m Further Zbelongs to  and since theequation | m-Fk-1 |=mand| Fk – m | = m cannot hold simultaneously. Zis smallerthan Z. This contradicts the definition Z again

Form 2, under the condition Fk m < Fk+1the theorem holds

REFERENCES [1] Pisot numbers and the Spectra of Real numbers –from the Ph.D. thesis of Kevin G. Hare.. [2] Kevin G. Hare, The Structure of the Spectra of Pisot numbers, 17 July 2003. [3] Erdos. I. Joo, and V.Komornik On the sequence ofnumbers of the form 0 +1q + ……+ nqn, i {0,1}.Acta.Arithmetica 83 (1998). 3.201 210. [4] B. Krishna Gandhi and H. Anita. “On Explorations of the Spectra of Pisot Numbers”. Indian Journal of Mathematics and Mathematical Sciences, Vol.2, No.1 (June 2006) 81-91.

1074 Dr. M Anita, Mr. G Nagendra Kumar, Ms. P Rajani