Generating the Silver Ratio  Using Different Approaches

 Naresh Patel*,  C.L. Parihar**

*Department of Applied Sciences (Mathematics), Institute of Engineering & Technology, Devi Ahilya University, Indore-452017(M.P.) India.

**Indian Academy of Mathematics, 15-Kaushaliyapuri, Chitawad Road, Indore-452001 (M.P.) India Email:




            Pell and Pell-Lucas numbers are respectively defined by the recurrence relation


            Their Binet formulae are     and    , where γ and δ                               are the roots of  x2 – 2x – 1 = 0 i.e.  and    so that                                    or           or         . Also          is called Silver ratio.

            In this paper, we have generated Silver ratio    using geometry, differential         equation, Newton-Raphson method, infinite continued fractions and bilinear   transformations. In the last, we have derived the formula for calculating number of digits in Pell and Pell-Lucas numbers, that is,  and . Some of the interesting results of this paper are as follows:

            (1)  where       Hence         , it follows that the sequence of        approximations               approaches     as .

(2) Taking  a triangle in which the square of one side is four times the product of other two sides, then the square root of the ratio of two sides of such triangle lies between  and   .

(3) The infinite continued fraction     converges to a limit    

            (4) The Bilinear transformation          has two fixed points             and                       if and           only if   , where a, b, c, d are integers; a, d > 0 and  ad-bc = 1.  

 (5)     .

(6) Number of digits in   = .

(7) Number of digits in =   

MSC 2000 Classification: 11B39

Key words: Silver Ratio, Pell and Pell-Lucas Numbers, Binet Formulae, Bilinear

Transformation, Number of Digits.



International eJournal of Mathematics and Engineering

Volume 2, Issue 2, Pages:  903 - 910