Almost Integer

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

Surprising examples are given by

(1)

which equals to within 5 digits and

(2)

which equals to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004). The first of these comes from the half-angle formula identity

(3)

where 22 is the numerator of the convergent 22/7 to , so . It therefore follows that any pi approximation gives a near-identity of the form .

Another surprising example involving both e and pi is

(4)

which can also be written as

(5) (6)

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

(7)

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" is true has yet been discovered.

Another nested cosine almost integer is given by

(8)

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

An example attributed to Ramanujan is

(9)

Some near-identities involving integers and the logarithm are

(10) (11) (12)

which are good to 6, 6, and 6 decimal digits, respectively (K. Hammond, pers. comm., Jan. 4 and Mar. 23-24, 2006).

An interesting near-identity is given by

(13)

(W. Dubuque, pers. comm.).

Near-identities involving and are given by

(14)

(D. Wilson, pers. comm.),

(15)

(D. Ehlke, pers. comm., Apr. 7, 2005),

(16)

(Povolotsky, pers. comm., May 11, 2008), and

(17)

(good to 8 digits; M. Stay, pers. comm., Mar. 17, 2009), or equivalently

(18)

Other remarkable near-identities are given by

(19)

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

(20)

(D. Davis, pers. comm.),

(21)

(posted to sci.math ; origin unknown),

(22)

(23)

(24)

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

(25)

(26)

(27) (28)

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

(29)

E. Pegg Jr. (pers. comm., Mar. 4, 2002) discovered the interesting near-identities

(30)

and

(31)

The near-identity

(32)

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

(33)

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

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

(34)

which is equal to 1 to five digits.

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

(35)

where is Khinchin's constant, as well as

(36)

(pers. comm., Feb. 4, 2005), where is the Euler-Mascheroni constant.

M. Joseph found

(37)

which is equal to 1 to four digits (pers. comm., May 18, 2006). M. Kobayashi (pers. comm., Sept. 17, 2004) found

(38)

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

(39)

which is equal to 0 to six digits (E. Pegg Jr., pers. comm., Sept. 28, 2004). S. M. Edde (pers. comm., Sep. 7, 2007) noted that

(40)

where is the digamma function.

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

(41) (42) (43)

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

(44)

(E. W. Weisstein, Feb. 5, 2005).

Prudnikov et al. (1986, p. 757) inadvertently give an almost integer result by incorrectly identifying the infinite product

(45)

where is a q-Pochhammer symbol, as being equal , which differs from the correct result by

(46)

A much more obscure almost identity related to the eight curve is the location of the jump in

(47)

where

(48)

and is an elliptic integral of the third kind, which is 1.3333292798..., or within of 4/3 (E. W. Weisstein, Apr. 2006). Another slightly obscure one is the value of needed to give a 99.5% confidence interval for a Student's t-distribution with sample size 30, which is 2.7499956..., or within of 11/4 (E. W. Weisstein, May 2, 2006).

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

(49)

which is within of .

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

(50)

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

(51)

for , as summarized in the following table.

0 0.72135 1 1.04068 2 3.00278 3 12.99629 4 74.99874 5 541.00152 6 4683.00125 7 47292.99873 8 545834.99791 9 7087261.00162 10 102247563.00527 11 1622632572.99755 12 28091567594.98157 13 526858348381.00125 14 10641342970443.08453 15 230283190977853.03744 16 5315654681981354.51308 17 130370767029135900.45799

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 people (where ties are allowed). Calling this number , it follows that

(52)

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

(53)

and then by contour integration it can be shown that

(54)

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

(55)

where is the Hurwitz zeta function. In fact, the other terms are quite small for from 1 to 15, so is the nearest integer to for these values, given by the sequence 1, 3, 13 75, 541, 4683, ... (OEIS 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 1988ab).

The properties of the j-function also give rise to the spectacular identity

(56)

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

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

25 37 43 58 67 74 148 0.00097 163 232 268 0.00029 522 652 719

Gosper (pers. comm.) noted that the expression

(57)

differs from an integer by a mere .

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

(58) (59)

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,

(60)

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

An example of a (very) near-integer is

(61) (62)

(Borwein and Borwein 1992; Maze and Minder 2005).

Maze and Minder (2005) found the class of near-identities obtained from

(63)

as

(64) (65) (66) (67) (68) (69) (70) (71)

(OEIS A114609 and A114610). Here, the excesses can be computed as exact sums connected by a recurrence relation, with the first few being

(72) (73)

(Maze and Minder 2005). These sums can also be done in closed form using q-polygamma functions , giving for example

(74) (75)

with .

An amusing almost integer involving units of length is given by

(76)

If combinations of physical and mathematical constants are allowed and taken in SI units, the following quantities have a near-integer numeric prefactor

(77) (78)

(M. Trott, pers. comm. Apr. 28, 2011), the first of which was apparently noticed by Weisskopf. Here, is the speed of light, is the elementary charge, is Boltzmann's constant, is Planck's constant, is the bond percolation threshold for a 4-dimensional hypercube lattice, is the vacuum permittivity, and is the Rydberg constant. Another famous example of this sort is Wyler's constant, which approximates the (dimensionless) fine structure constant in terms of fundamental mathematical constants.