**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 *φ*_{n} as using n = {5, 6, 7} terms respectively, then,

(1/2)(*φ*-*φ*_{5})(2*φ*)^{5} = 1.0977…

(1/2)(*φ*-*φ*_{6})(2*φ*)^{6} = 1.0983…

(1/2)(*φ*-*φ*_{7})(2*φ*)^{7} = 1.0985…

which is approaching the Paris constant *R* = 1.09864196… It seems this result can be generalized to radicals of *k*th 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* = *x*_{n} 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-x_{n})(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 x^{3} = x+1. Since y = 1/P+1 = 1.75488…, then for small *n*,

(P-P_{5})(3y)^{5} = 1.81673…

(P-P_{6})(3y)^{6} = 1.81685…

(P-P_{7})(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_{(1,10}^{14}_{)} = 1.3862943611198999…

C_{(1,10}^{15}_{)} = 1.3862943611198915…

C_{(1,10}^{16}_{)} = 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*