π in terms of φ via the Machin s Route

Transcription

1 in terms of φ via the Machin s Route Hei-Chi Chan Mathematical Science Program, University of Illinois at Springfield Springfield, IL chan.hei-chi@uis.edu Abstract In this paper, we prove some formulas for that are expressed in terms of the powers of the reciprocal of the Golden Ratio φ. These formulas depend on Machin-type identities like the following: =2arctan φ 3 + arctan φ 5. Introduction In a recent paper [8], we proved several formulas of which are expressed in terms of the reciprocal of the Golden Ratio φ; for example = 5 2+φ 2 φ n=0 5n 2φ 5n (5n +2) 2 2 φ 3 (5n +3) 2 3 φ 3 (5n +) () These formulas were inspired by the work of Bailey, Borwein and Plouffe (BBP) [6], who proved a family of amazing formulas for with the aid of the powerful PSLQ

2 algorithm [9]. As an example, they proved that = n=0 ( 6 n 8n + 2 8n + 8n +5 ). (2) 8n +6 For an introduction and generalizations of (2), see, e.g., [, 2, 7]; see also the lucid account in Hijab s book [20]. For a compendium of currently known results of BBPtype formulas, see Bailey s A Compendium of BBP-Type Formulas for Mathematical Constants, which is available at dhbailey. In this paper, we prove several formulas for that share similarities with (): they express in terms of the reciprocal of the Golden Ratio φ: = ( ) k 2k + 2k+ + φ ( ) k 2k+, (3) 2k + φ 3 = 2 = 3 ( ) k 2k+ + 2k + ( ) k 2k+ + 2k + φ 3 ( ) k 2k+, () 2k + φ 6 ( ) k 2k+. (5) 2k + φ 5 The derivations of these formulas (cf. the next section) are analogous to the formula discovered by Machin ( ). Machin s discovery starts with the following observation: = arctan arctan. (6) By applying to (6) the power series of arctan x, i.e., arctan x = we have the Machin s formula for : ( ) k x2k+ 2k +, (7) = ( ) k 2k + 2k+ 5 2 ( ) k 2k+. (8) 2k + 239

3 Machin s discovery plays a key role in computing the digits of, see [7]. See also [8, 3]. For generalizations of Machin s formula, see [5, 7]. See also Weisstein s article [22]. It should be note that there are Machin-Type formulas that are closely related to the Fibonacci numbers. For example, we have = k= arctan. F 2k+ Cf. chapter 2 of Koshy s book [2]. See also Ron Knott s award-winning webstie at We also remark that in an interesting paper [], Jon Borwein, David Borwein and William Galway explored a class of Machin-type BBP formulas. Next, we will turn to the proof of (3)-(5). 2 Proofs of the Main Formulas To prove (3)-(5), all we need to do is to establish the following identities: = arctan + arctan, (9) φ φ 3 = 2 arctan + arctan, (0) φ 6 = 3 arctan + arctan. () φ 3 φ 5 By using (7), it follows at once that (9) leads to (3), (0) to () and () to (5). We will say a few words on how these identities were discovered in the next section. 3

4 First, we recall two identities that will be useful in our proof: for n 2 φ n = F n φ + F n (2) and for n, φ n =( ) n F n φ +( ) n F n+ (3) Here, F n is the n th Fibonnaci number. For proofs, see p. 78 of [2], or p. 38 in [2]. The latter is based on probablistic approach; see also the papers [9, 0, ]. Let us define γ = arctan (/φ 3 ). In order to prove (), we need to show that ( ) tan 3γ = φ. () 5 First, we claim that Indeed, we have tan(2γ) = 2. (5) tan 2γ = = = 2tanγ tan 2 γ = 2φ 3 φ 6 2 φ 3 φ 3 2 F 3 φ + F 2 (F 3 φ F ) = 2. Note that we have used (2) and (3) for φ ±3 to derive the third equality. Next, we show that in a similar manner Indeed, by using (5) for tan 2γ we have tan 3γ = tan(3γ) = 3+2φ +φ. (6) tan γ +tan2γ tan γ tan 2γ = 2+φ3 2φ 3 = 3+2φ +φ. Note that we have used (2) and (3) to obtain the last equality.

5 With (6), we can establish (): ( ) tan 3γ = tan 3γ +tan3γ = φ 3φ +2 = φ φ = φ. 5 In the third equality, we have used φ =φ (i.e., n = in (3)) to rewrite the numerator, and 3φ +2=φ (i.e., n = in (2)) to rewrite the denominator. This establishes () and implies (5). The proofs of the other two equations, namely, (9) and (0), follow the same pattern and we briefly comment on these proofs. For (9), let us define α = arctan (/φ). Then, ( ) tan α = tan α +tanα = φ φ + = φ φ = 2 φ. 3 Note that we have used the fact that φ =φ (i.e., n = in (3)) and = φ + (i.e., n = 2 in (2)). This proves (9). For (0), let us define β = arctan (/ ). Then, we have tan 2β = 2tanβ tan 2 β = 2 φ 2 = 2 2φ. Note that we have used (2) and (3) to obtain the last line. Finally, ( ) tan 2β = tan 2β +tan2β = 2φ 3 2φ + = φ 3 φ = 3 φ. 6 This proves (0). As a by-product, we note that (5) implies the following identity arctan = φ 3 2 arctan. 2 3 Looking Back and Ahead Originally, identities (9) to () were discovered in a series of numerical experiments using Mathematica. The following is an account of how we discovered (0). 5

6 Motivated by Machin formula (6), we first set out to look for an identity of the following form: = A arctan + B arctan. (7) φ k Our goal was to determine constants A, B and k. Next, we treated the first term in (7) as a first order approximation of /: A arctan. We compared this approximation with the following numerical result 2.5 arctan. This motivated us to set A = 2 in (7): = 2 arctan + B arctan. (8) φ k We then looked for B and k that would make (8) an identity. We performed a series of numerical experiments for this purpose. Precisely, we used Mathematica to compute, for a range of k, the following ratio which represents B (cf. (8)): 2 arctan φ 2. arctan φ k Gladly, we found that at k = 6 (a relatively small k!), Mathematica computed the ratio to be (where we requested the software to give the first 30 decimal places). This suggested that we should try to prove or disprove the identity? = 2 arctan + arctan. (9) φ 6 6

7 With joy and gratitude, we found that (9) turned out to be an exact identity. One can generalize the present work by considering recursions with three or more terms. See [22]. We believe that more sophisticated numerical experiments, along the overall theme suggested in [5, 6], may point to more discoveries of this type of identities. See also [3,, 5]. Acknowledgment. I would like to thank Scott Ebbing for his valuable comments on this work. References [] V. Adamchik and S. Wagon, : A 2000-Year Search Changes Direction, Mathematica in Education and Research (996), 9. [2] V. Adamchik and S. Wagon, A Simple Formula for Pi, Amer. Math. Monthly 0 (997), [3] D. H. Bailey and J. M. Borwein, Experimental Mathematics: Recent Developments and Future Outlook, in Mathematics Unlimited 200 and Beyond, Bjorn Engquist and Wilfried Schmid, ed., Springer, New York, 200, [] D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices Amer. Math. Soc. 52 (2005), [5] D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics, MAA Monthly to appear. [6] D. H. Bailey, P. B. Borwein and S. Plouffe, On the Rapid Computation of Various Polylogarithmic Constants, Math. Comp. 66 (997),

8 [7] D. H. Bailey and S. Plouffe, Recognizing Numerical Constants, in The Organic Mathematics Project Proceedings, April 2, 996; hard copy version: Canadian Mathematical Society Conference Proceedings 20 (997) [8] P. Beckmann, The History of, 3 rd ed., St. Martin s Griffin, New York, 976. [9] A. T. Benjamin, C. R.H. Hanusa and F. E. Su, Linear Recurrences Through Tilings and Markov Chains Utilitas Mathematica 6 (2003), 3 7. [0] A. T. Benjamin, G. M. Levin, K. Mahlburg and J. J. Quinn, Random Approaches to Fibonacci Identities Amer. Math. Monthly 07 (2000), [] A. T. Benjamin, J. D. Neer, D. E. Otero and J. A. Sellers, A Probabilistic View of Certain Weighted Fibonacci Sums The Fibonacci Quart. (2003), [2] A. T. Benjamin and J. J. Quinn, Proofs that Really Count: the Art of Combinatorial Proof, MAA, Washington, [3] D. Blatner, The Joy of, Walker and Company, New York, 997. [] D. Borwein, J. M. Borwein, and W. F. Galway, Finding and Excluding -ary Machin-Type BBP Formulae, Canadian J. Math 56 (200), [5] J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 2st Century, A K Peters, Massachusetts, 200. [6] J. M. Borwein, D. H. Bailey, R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Massachusetts,

9 [7] J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, CMS Series of Monographs and Advanced books in Mathematics, John Wiley, New Jersey, 987. [8] H. C. Chan, in terms of φ, Fibonacci Quart., to appear. [9] H. R. P. Ferguson, D. H. Bailey and S. Arno, Analysis of PSLQ, An Integer Relation Finding Algorithm, Mathematics of Computation 68 (999), [20] O. Hijab, Introduction to Calculus and Classical Analysis, Springer-Verlag, New York, 997. [2] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley, New York, 200. [22] E. W. Weisstein. Machin-Like Formulas. From MathWorld A Wolfram Web Resource. AMS Subject Classification: A05 9