### 0014: Article 4 (Golden Ratio and Nested Radicals)

 Back to Index       Go to Updates Page     Have You Seen This Number N = 1.8168834242447…?   By Tito Piezas III     Abstract:  “We consider the asymptotic behavior of the infinite nested radical defining the golden ratio φ and plastic constant P,       and give a conjecture on the general case.”       I. Introduction   While trying to find more commonalities between the golden ratio φ and plastic constant (see article, A Tale of Four Constants), this author came across the Paris constant.  In 1987, R. Paris proved that the nested radical expression for φ given above approaches φ at a constant rate.  For example, defining φn as using n = {5, 6, 7} terms respectively, then,   (1/2)(φ-φ5)(2φ)5 = 1.0977… (1/2)(φ-φ6)(2φ)6 = 1.0983… (1/2)(φ-φ7)(2φ)7 = 1.0985…   which is approaching the Paris constant R = 1.09864196… It seems this result can be generalized to radicals of kth power, hence includes the plastic constant and other values.   Conjecture 1:  “Given the infinite nested radical,     for integer k > 1, non-zero a, and the equation (eq.1),     Let x be the root of eq.1 such that x = xn as n → ∞.  Define,     Then,     for some constant C(a,k).”     II. Particular cases   For quadratic radicals k = 2, since y = a/x + 1 = x, then this is simply,   (x-xn)(2x)n → C(a,2)   which apparently is relatively easy to prove.  The special case a = 1,     yields C(1,2) = 2R = 2.19728392…, with the Paris constant R = 1.09864196… (R. Paris, An Asymptotic Approximation Connected with the Golden Number, Amer. Math. Monthly, 1987).  For a = 2,     as Denis Feldmann points out, the constant C(a,k) has a nice closed-form expression as,   C(2,2) = (π/2)2 = 2.46740110…   For cubic radicals, say, {a,k} = {1,3}, we get the plastic constant ,     where P = 1.32472…, the real root of x3 = x+1.  Since y = 1/P+1 = 1.75488…, then for small n,   (P-P5)(3y)5 = 1.81673… (P-P6)(3y)6 = 1.81685… (P-P7)(3y)7 = 1.81687…   As n increases, one can see this is approaching the constant,   C(1,3) = 1.8168834242447…   mentioned in the title and accurate to 12 decimal places.  And so on for other values {a,k}.  Note, however, that for the special case a = 1,   C(1,2) = 2.19728… C(1,3) = 1.81688… C(1,4) = 1.68029… . . . C(1,152) = 1.39246… C(1,153) = 1.39242… C(1,154) = 1.39238…   Conjecture 2:  “The sequence of constants C(a,k) for a = 1 as k → ∞ is also approaching a constant.”   Question 1:  Anybody knows how to prove/disprove Conjectures 1 and 2?  If indeed Conjecture 2 is true, what is this constant, or, at least, what are its first dozen or so decimal digits?   (Update, 10/4/10): Courtesy of Adam Strzebonski and Daniel Lichtblau of Wolfram Research, I managed to find C(1,k) for very high k.  The convergence is VERY slow, taking powers of 10 to barely move one decimal digit:   C(1,1014) = 1.3862943611198999… C(1,1015) = 1.3862943611198915… C(1,1016) = 1.3862943611198907…   Using Plouffe’s Inverse Symbolic Calculator, one finds that,   2 ln(2)  =  1.3862943611198906…   so it is a reasonable conjecture that the sequence of constants C(1,k) as k → ∞ has its limit 2 ln(2).  Note that the expression 2 ln(2) also appears in Lévy’s constant γ,     another constant involved in asymptotic behavior, this time for the denominators of the convergents of continued fractions.  (End update)   Question 2:  Is there a closed-form expression for C(a,k)  for other values {a,k} ?     -- End --     P.S. The next article will be on Ramanujan's beautiful continued fractions and its interesting connection to Platonic solids.       © Oct 2010, Tito Piezas III You can email author at tpiezas@gmail.com        ◄