Ramanujan Constant

The Irrational constant

$$R\equiv e^{\pi\sqrt{163}}=262537412640768743.99999999999925\ldots$$

which is very close to an Integer. Numbers such as the Ramanujan constant can be found using the theory of Modular Functions. A few rather spectacular examples are given by Ramanujan (1913-14), including the one above, and can be generated using some amazing properties of the j-Function.

M. Gardner (Apr. 1975) played an April Fool's joke on the readers of Scientific American by claiming that this number was exactly an Integer. He admitted the hoax a few months later (Gardner, July 1975).

See also Almost Integer, Class Number, j-Function

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 387, 1987.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Gardner, M. ``Mathematical Games: Six Sensational Discoveries that Somehow or Another have Escaped Public Attention.'' Sci. Amer. 232, 127-131, Apr. 1975.

Gardner, M. ``Mathematical Games: On Tessellating the Plane with Convex Polygons.'' Sci. Amer. 232, 112-117, Jul. 1975.

Good, I. J. ``What is the Most Amazing Approximate Integer in the Universe?'' Pi Mu Epsilon J. 5, 314-315, 1972.

Plouffe, S. `` $e^{\pi\sqrt{163}}$, the Ramanujan Number.'' http://www.lacim.uqam.ca/piDATA/ramanujan.txt.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.

Wolfram, S. The Mathematica Book, 3rd ed. New York: Cambridge University Press, p. 52, 1996.