(OEIS A060295), which is an almost integer. Numbers such as the Ramanujan constant can be found using the theory
of modular functions. In fact, the nine Heegner
numbers (which include 163) share a deep number theoretic property related to
some amazing properties of the j-function that
leads to this sort of near-identity.
Although Ramanujan (1913-1914) gave a few rather spectacular examples of almost integers (such as ), he did not actually mention the particular
near-identity given above. In fact, Hermite (1859) observed this property of 163
long before Ramanujan's work. The name "Ramanujan's constant" was coined
by Simon Plouffe (Michon 2003) and derives from an April Fool's joke played by Martin
Gardner (1975a) on the readers of Scientific American. In his column, Gardner
claimed that
was exactly an integer, and that Ramanujan had conjectured
this in his 1914 paper. Gardner admitted his hoax a few months later (Gardner 1975b).
An approximation that agrees with the Ramanujan constant through 14 digits
after the decimal point is
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, 126-133, Apr.
1975a. https://doi.org/10.1038/scientificamerican0475-126.Gardner,
M. "Mathematical Games: On Tessellating the Plane with Convex Polygons."
Sci. Amer.233, 112-117, Jul. 1975b. https://doi.org/10.1038/scientificamerican0775-112.Good,
I. J. "What Is the Most Amazing Approximate Integer in the Universe?"
Pi Mu Epsilon J.5, 314-315, 1972.Hermite, C. "Sur
la théorie des équations modulaires." C. R. Acad. Sci.
Paris49, 16-24, 110-118, and 141-144, 1859. Reprinted in Oeuvres complètes,
Tome II. Paris, France: Hermann, p. 61, 1912.Michon, G. P.
"Final Answers: Numerical Constants." Numericana. Aug. 6, 2003.
https://www.numericana.com/answer/constants.htm#ramanju.Numberphile.
"163 and Ramanujan Constant." Mar. 2, 2012. https://www.youtube.com/watch?v=DRxAVA6gYMM.Piezas,
T. III "Ramanujan's Constant and Its Cousins." Jan. 14, 2005. https://www.oocities.org/titus_piezas/Ramanujan_a.htm.Ramanujan,
S. "Modular Equations and Approximations to ." Quart. J. Pure Appl. Math.45, 350-372,
1913-1914.Sloane, N. J. A. Sequences A060295
and A102912 in "The On-Line Encyclopedia
of Integer Sequences."Wolfram, S. A
New Kind of Science. Champaign, IL: Wolfram Media, p. 913,
2002.Wolfram, S. The
Mathematica Book, 5th ed. Champaign, IL: Wolfram Media, p. 33, 2003.Yui,
N. and Zagier, D. "On the Singular Values of Weber Modular Functions."
Math. Comput.66, 1645-1662, 1997. https://doi.org/10.1090/S0025-5718-97-00854-5.
), he did not actually mention the particular
near-identity given above. In fact, Hermite (1859) observed this property of 163
long before Ramanujan's work. The name "Ramanujan's constant" was coined
by Simon Plouffe (Michon 2003) and derives from an April Fool's joke played by Martin
Gardner (1975a) on the readers of Scientific American. In his column, Gardner
claimed that 
was exactly an integer, and that Ramanujan had conjectured
this in his 1914 paper. Gardner admitted his hoax a few months later (Gardner 1975b).
denotes the unique real polynomial root of 
.
is a value of the Weber modular function
(Yui and Zagier 1997, pp. 1660-1661). The infinite
product for 
gives
, corresponding to a relative
error of 
.
." Quart. J. Pure Appl. Math. 45, 350-372,
1913-1914.Sloane, N. J. A. Sequences