Almost Integer

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

$$e^\pi-\pi=19.999099979\ldots,$$ (1)

which can also be written as

$$(\pi+20)^i = -0.9999999992-0.0000388927i \approx -1$$ (2)

$$\cos(\ln(\pi + 20)) \approx -0.9999999992.$$ (3)

Applying Cosine a few more times gives

$$\cos(\pi\cos(\pi\cos(\ln(\pi+20))))\approx -1+3.9321609261\times 10^{-35}.$$ (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

$${1\over 4}\left[{\cos({\textstyle{1\over 10}})+\cosh({\texts... ...le{1\over 20}}\sqrt{2}\,)}\right]=1+2.480\ldots\times 10^{-13}$$ (5)

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

$${5(1+\sqrt{5}\,)[\Gamma({\textstyle{3\over 4}})]^2\over e^{5\pi/6}\sqrt{\pi}}=1+4.5422\ldots\times 10^{-14},$$ (6)

where $\Gamma(z)$ is the Gamma Function (S. Plouffe), and

$$e^6-\pi^4-\pi^5=0.000017673\ldots$$ (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 $e^{\pi\sqrt{163}}$ (sometimes known as the Ramanujan Constant and which corresponds to the field $\Bbb{Q}(\sqrt{-163}\,)$ which has Class Number 1 and is the Imaginary quadratic field of maximal discriminant), $e^{\pi\sqrt{22}}$, $e^{\pi\sqrt{37}}$, and $e^{\pi\sqrt{58}}$, 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

$$\left[{\ln(640320^3+744)\over\pi}\right]^2=163+2.32167\ldots\times 10^{-29}$$ (8)

(Le Lionnais 1983, p. 152).

The list below gives numbers of the form $x\equiv e^{\pi\sqrt{n}}$ for $n\leq 1000$ for which $\left\lceil{x}\right\rceil -x \leq 0.01$.

$\quad e^{\pi\sqrt{6}} = 2,197.990\,869\,543\ldots$
$\quad e^{\pi\sqrt{17}} = 422,150.997\,675\,680\ldots$
$\quad e^{\pi\sqrt{18}} = 614,551.992\,885\,619\ldots$
$\quad e^{\pi\sqrt{22}} = 2,508,951.998\,257\,424\ldots$
$\quad e^{\pi\sqrt{25}} = 6,635,623.999\,341\,134\ldots$
$\quad e^{\pi\sqrt{37}} = 199,148,647.999\,978\,046\,551\ldots$
$\quad e^{\pi\sqrt{43}} = 884,736,743.999\,777\,466\ldots$
$\quad e^{\pi\sqrt{58}} = 24,591,257,751.999\,999\,822\,213\ldots$
$\quad e^{\pi\sqrt{59}} = 30,197,683,486.993\,182\,260\ldots$
$\quad e^{\pi\sqrt{67}} = 147,197,952,743.999\,998\,662\,454\ldots$
$\quad e^{\pi\sqrt{74}} = 545,518,122,089.999\,174\,678\,853\ldots$
$\quad e^{\pi\sqrt{149}} = 45,116,546,012,289,599.991\,830\,287\ldots$
$\quad e^{\pi\sqrt{163}} = 262,537,412,640,768,743.999\,999\,999\,999\,250\,072\ldots$
$\quad e^{\pi\sqrt{177}} = 1,418,556,986,635,586,485.996\,179\,355\ldots$
$\quad e^{\pi\sqrt{232}} = 604,729,957,825,300,084,759.999\,992\,171\,526\ldots$
$\quad e^{\pi\sqrt{267}} = 19,683,091,854,079,461,001,445.992\,737\,040\ldots$
$\quad e^{\pi\sqrt{326}} = 4,309,793,301,730,386,363,005,719.996\,011\,651\ldots$

$\quad e^{\pi\sqrt{386}} = 639,355,180,631,208,421,212,174,016.997\,669\,832\ldots$
$\quad e^{\pi\sqrt{522}} = 14,871,070,263,238,043,663,567,627,879,007.999\,848\,726\ldots$
$\quad e^{\pi\sqrt{566}} = 288,099,755,064,053,264,917,867,975,825,573.993\,898\,311\ldots$
$\quad e^{\pi\sqrt{638}} = 28,994,858,898,043,231,996,779,771,804,797,161.992\,372\,939\ldots$
$\quad e^{\pi\sqrt{719}} = 3,842,614,373,539,548,891,490,294,277,805,829,192.999\,987\,249\ldots$
$\quad e^{\pi\sqrt{790}} = 223,070,667,213,077,889,794,379,623,183,838,336,437.992\,055\,117\ldots$
$\quad e^{\pi\sqrt{792}} = 249,433,117,287,892,229,255,125,388,685,911,710,805.996\,097\,323\ldots$
$\quad e^{\pi\sqrt{928}} = 365,698,321,891,389,219,219,142,531,076,638,716,362,775.998\,259\,747\ldots$
$\quad e^{\pi\sqrt{986}} = 6,954,830,200,814,801,770,418,837,940,281,460,320,666,108.994\,649\,611\ldots.$

Gosper noted that the expression

$1 - 262537412640768744 e^{-\pi\sqrt{163}}- 196884 e^{-2\pi\sqrt{163}}$
$+ 103378831900730205293632 e^{-3\pi\sqrt{163}}.\quad$	(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 $\pi$.'' 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.