## Almost Integer

A number which is very close to an Integer. One surprising example involving both e and Pi is

 (1)

which can also be written as
 (2)

 (3)

Applying Cosine a few more times gives

 (4)

This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to why'' it has been true has yet been discovered.

An interesting near-identity is given by

 (5)

(W. Dubuque). Other remarkable near-identities are given by
 (6)

where is the Gamma Function (S. Plouffe), and
 (7)

(D. Wilson).

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)

(Le Lionnais 1983, p. 152).

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.

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.