An almost integer is a number that is very close to an integer.

Surprising examples are given by

which equals -1 to within 5 digits and

which equals -1 to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004).

Another surprising example involving both e and pi is

which can also be written as

Here, is Gelfond's constant. Applying cosine a few more times gives

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.

Another nested cosine almost integer is given by

(P. Rolli, pers. comm., Feb. 19, 2004).

An example attributed to Ramanujan is

An interesting near-identity is given by

(W. Dubuque, pers. comm.).

Other remarkable near-identities are given by

where is the gamma function (S. Plouffe, pers. comm.),

(D. Wilson, pers. comm.),

where is the root of (L. A. Broukhis, pers. comm.),

(D. Davis, pers. comm.),

(posted to sci.math; origin unknown),

where C is Catalan's constant, is the Euler-Mascheroni constant, and is the golden ratio (D. Barron, pers. comm.), and

(E. Stoschek, pers. comm.). Stoschek also gives an interesting near-identity involving the fine structure constant and Feigenbaum constant ,

E. Pegg (pers. comm.) discovered the interesting near-identities

and

(March 4, 2002). The near-identity

arises by noting that the stellation ratio in the cumulation of the dodecahedron to form the great dodecahedron is approximately equal to . Another near identity is given by

where is Apéry's constant and is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).

M. Hudson (pers. comm., Oct. 18, 2004) noted the almost integer

where K is Khinchin's constant.

J. DePompeo (pers. comm., Mar. 29, 2004) found

which is equal to 1 to five digits.

M. Kobayashi (pers. comm., Sept. 17, 2004) found

which is equal to 1 to five digits. The related expression

which is equal to 0 to six digits (E. Pegg Jr., pers. comm., Sept. 28, 2004).

E. W. Weisstein (Mar. 17, 2003) found the almost integers

as individual integrals in the decomposition of the integration region to compute the average area of a triangle in triangle triangle picking.

and give the almost integer

(E. W. Weisstein, Feb. 5, 2005).

Let be the average length of a line in triangle line picking for an isosceles right triangle, then

which is within of .

D. Terr (pers. comm., July 29, 2004) found the almost integer

where is the golden ratio and is the natural logarithm of 2.

A set of almost integers due to D. Hickerson are those of the form

for , as summarized in the following table.

These numbers are close to integers due to the fact that the quotient is the dominant term in an infinite series for the number of possible outcomes of a race between n people (where ties are allowed). Calling this number f(n), it follows that

with , where is a binomial coefficient. From this, we obtain the exponential generating function for f

and then by contour integration it can be shown that

for , where i is the square root of -1 and the sum is over all integers k (here, the imaginary parts of the terms for k and cancel each other, so this sum is real). The k = 0 term dominates, so f(n) is asymptotic to . The sum can be done explicitly as

where is the Hurwitz zeta function. In fact, the other terms are quite small for n from 1 to 15, so f(n) is the nearest integer to for these values (Hickerson), given by the sequence 1, 3, 13 75, 541, 4683, ... (Sloane's A034172).

A large 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

(Le Lionnais 1983, p. 152; Trott 2004, p. 8).

The list below gives numbers of the form for for which .

Gosper (pers. comm.) noted that the expression

	(42)

differs from an integer by a mere .

E. Pegg Jr. noted that the triangle dissection illustrated above has length

which is almost an integer.

Borwein and Borwein (1992) and Borwein et al. (2004, pp. 11-15) give examples of series identities that are nearly true. For example,

which is true since and for positive integer n < 268. In fact, the first few doubled values of n at which are 268, 536, 804, 1072, 1341, 1609, ... (Sloane's A096613).

Class Number, Floor, j-Function, Pi, Pisot Number, Triangle Dissection