Grossman's Constant


Define the sequence a_0=1, a_1=x, and


for n>=0. The first few values are


Janssen and Tjaden (1987) showed that this sequence converges for exactly one value x=c, where c=0.73733830336929... (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of a_n for n=1 to 30, with odd n shown in red and even n shown in blue, for x ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed x<c, the red values go to 0, while the blue values go to some positive number.

Nyerges (2000) has generalized the recurrence to the functional equation


See also

Foias Constant

Explore with Wolfram|Alpha


Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse and G. Păun). London: Springer-Verlag, pp. 119-126, 2000.Finch, S. R. "Grossman's Constant." §6.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 429-430, 2003.Grossman, J. W. "Problem 86-2." Math. Intel. 8, 31, 1986.Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Problem 86-2. Math. Intel. 9, 40-43, 1987.Michon, G. P. "Final Answers: Numerical Constants.", G. "The Solution of the Functional Equation x=(1+F(x))F^2(x)." Preprint, Oct. 19, 2000., N. J. A. Sequence A085835 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Grossman's Constant

Cite this as:

Weisstein, Eric W. "Grossman's Constant." From MathWorld--A Wolfram Web Resource.

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