A number which is very close to an Integer. One surprising example involving both e and Pi is
(1) |
(2) |
(3) |
(4) |
An interesting near-identity is given by
(5) |
(6) |
(7) |
A whole class of Irrational ``almost integers'' can be found using the theory of Modular Functions, and a few rather spectacular examples are given by Ramanujan (1913-14). Such approximations were also studied by Hermite (1859), Kronecker (1863), and Smith (1965). They can be generated using some amazing (and very deep) properties of the j-Function. Some of the numbers which are closest approximations to Integers are (sometimes known as the Ramanujan Constant and which corresponds to the field which has Class Number 1 and is the Imaginary quadratic field of maximal discriminant), , , and , the last three of which have Class Number 2 and are due to Ramanujan (Berndt 1994, Waldschmidt 1988).
The properties of the j-Function also give rise to the spectacular identity
(8) |
The list below gives numbers of the form for for which .
Gosper noted that the expression
(9) |
See also Class Number, j-Function, Pi
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 90-91, 1994.
Hermite, C. ``Sur la théorie des équations modulaires.'' C. R. Acad. Sci. (Paris) 48, 1079-1084 and 1095-1102, 1859.
Hermite, C. ``Sur la théorie des équations modulaires.'' C. R. Acad. Sci. (Paris) 49, 16-24, 110-118, and 141-144, 1859.
Kronecker, L. ``Über die Klassenzahl der aus Werzeln der Einheit gebildeten komplexen Zahlen.'' Monatsber. K. Preuss. Akad. Wiss. Berlin, 340-345. 1863.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.
Ramanujan, S. ``Modular Equations and Approximations to .'' Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.
Smith, H. J. S. Report on the Theory of Numbers. New York: Chelsea, 1965.
Waldschmidt, M. ``Some Transcendental Aspects of Ramanujan's Work.'' In
Ramanujan Revisited: Proceedings of the Centenary Conference (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin).
New York: Academic Press, pp. 57-76, 1988.
© 1996-9 Eric W. Weisstein