Almost integer

In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one.

Almost integers may be considered interesting when they arise in some context in which they are unexpected.

Some examples of almost integers are high powers of the golden ratio

ϕ =

, for example: The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number.

The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision: As n increases, the number of consecutive nines or zeros beginning at the tenths place of a(n) approaches infinity.

Other occurrences of non-coincidental near-integers involve the three largest Heegner numbers: where the non-coincidence can be better appreciated when expressed in the common simple form:[2] where and the reason for the squares is due to certain Eisenstein series.

The constant

is sometimes referred to as Ramanujan's constant.

Almost integers that involve the mathematical constants π and e have often puzzled mathematicians.

The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows:

The first term dominates since the sum of the terms for

The sum can therefore be truncated to

Rewriting the approximation for

Thus, rearranging terms gives

Ironically, the crude approximation for

yields an additional order of magnitude of precision.

[1] Another example involving these constants is: