Almost integer

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In recreational mathematics an almost integer is an irrational number that is surprisingly close to an integer. Well known examples of almost integers are high powers of the golden ratio $\scriptstyle \varphi\, =\, 1.61803\,39887\dots$ , e.g.:

$\varphi^{17} = 3571.000280\dots\,$

$\varphi^{18} = 5777.999827\dots\,$

$\varphi^{19} = 9349.000107\dots\,$

The fact that these powers approach integers is non-coincidental, and is related to the fact that the golden ratio is a Pisot-Vijayaraghavan number: an algebraic integer with conjugate elements that are in absolute value smaller than unity. It follows that for $\scriptstyle n \,\gg\, 1\,$ :

$\varphi^n \approx L_n - \frac{(-1)^n}{L_n}$

where L_n is the nth Lucas number.

Other occurrences of non-coincidental near-integers involve the three largest Heegner numbers:

$e^{\pi \sqrt{43}} = 884736743.99977\dots\,.$

$e^{\pi \sqrt{67}} = 147197952743.9999986\dots\,.$

$e^{\pi \sqrt{163}}= 262537412640768743.99999999999925\dots\,.$

where the non-coincidence can be better appreciated when expressed in the common simple form^[1]:

$\begin{align} e^{\pi \sqrt{43}} &\approx 12^3(9^2-1)^3+744-.00022\\ e^{\pi \sqrt{67}} &\approx 12^3(21^2-1)^3+744-.0000013\\ e^{\pi \sqrt{163}} &\approx 12^3(231^2-1)^3+744-.00000000000075 \end{align}$

and the reason for the squares being due to certain Eisenstein series. The constant $\scriptstyle e^{\pi \sqrt{163}}$ is sometimes referred to as Ramanujan's constant.

Almost integers involving the mathematical constants pi and e have often puzzled mathematicians. An example is

$e^\pi - \pi = 19.999099979\dots\,.$

To date, no explanation has been given for this fact why Gelfond's constant is nearly identical to $\scriptstyle \pi \,+\, 20$ ,^[2] which is therefore regarded to be a mathematical coincidence.