| 释义 |
Almost IntegerA 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) |
 is the Gamma Function (S. Plouffe), and
 | (7) |
A whole class of Irrational ``almost integers'' can be found using the theory ofRamanujan  (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 thenumbers which are closest approximations to Integers are  (sometimes known as theRamanujan Constant and which corresponds to the field  which has Class Number 1 and is theImaginary quadratic field of maximal discriminant),  ,  ,and  , the last three of which have Ramanujan (Berndt1994, 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) |
differs from an Integer by a mere 10-59.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. |
