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 are considered interesting when they arise in some context in which they are unexpected.

## Almost integers related to e and π

• ${\displaystyle e^{\pi\sqrt{37}}=20836851\mathcal{E}.\mathcal{E}\mathcal{E}\mathcal{E}7476257520\mathcal{E}483574\mathcal{X}86\mathcal{X}\ldots}$
• ${\displaystyle e^{\pi\sqrt{57}}=2464036851\mathcal{E}.\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}800\mathcal{X}683617\mathcal{E}91255\mathcal{E}09\ldots}$
• ${\displaystyle e^{\pi\sqrt{117}}=1505939952036851\mathcal{E}.\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}\mathcal{E}5391\mathcal{X}2899\mathcal{X}944\ldots}$

(rounded to 20 significant figures after the dozenal point)

They are the numbers ${\displaystyle e^{\pi\sqrt{n}}}$, where n are one of the largest three Heegner numbers (37, 57 and 117).

The forms of them are

• ${\displaystyle e^{\pi\sqrt{37}}=680^3+520-4.74596469\mathcal{E}07\ldots\times 10^{-4}}$
• ${\displaystyle e^{\pi\sqrt{57}}=3080^3+520-3.\mathcal{E}\mathcal{E}15385\mathcal{X}402\ldots\times 10^{-6}}$
• ${\displaystyle e^{\pi\sqrt{117}}=26\mathcal{X}680^3+520-6.82\mathcal{X}19322127\ldots\times 10^{-10}}$

Alternatively,

• ${\displaystyle e^{\pi\sqrt{37}}=10^3(9^2-1)^3+520-4.74596469\mathcal{E}07\ldots\times 10^{-4}}$
• ${\displaystyle e^{\pi\sqrt{57}}=10^3(19^2-1)^3+520-3.\mathcal{E}\mathcal{E}15385\mathcal{X}402\ldots\times 10^{-6}}$
• ${\displaystyle e^{\pi\sqrt{117}}=10^3(173^2-1)^3+520-6.82\mathcal{X}19322127\ldots\times 10^{-10}}$

(the numbers with scientific notation are rounded to 10 significant figures)

where the reason for the squares is due to certain Eisenstein series. For Heegner numbers ${\displaystyle d < 17}$, one does not obtain an almost integer; even ${\displaystyle d = 17}$ is not noteworthy (because of d = 17, the form is ${\displaystyle e^{\pi\sqrt{17}}=80^3+520-0.280203603728\ldots}$, but the absolute deviation of a random real number (picked uniformly from ${\displaystyle [0,1]}$, say) is a uniformly distributed variable on ${\displaystyle [0,0.6]}$, so it has absolute average deviation and median absolute deviation of 0.3, and a deviation of 0.28 is not exceptional.). The integer j-invariants are highly factorisable, which follows from the ${\displaystyle 10^3(n^2-1)^3=(2^2\cdot 3 \cdot (n-1) \cdot (n+1))^3}$ form, and factor as,

• ${\displaystyle j((1+\sqrt{-37})/2)=680^3=(2^6 \cdot 3 \cdot 5)^3}$
• ${\displaystyle j((1+\sqrt{-57})/2)=3080^3=(2^5 \cdot 3 \cdot 5 \cdot \mathcal{E})^3}$
• ${\displaystyle j((1+\sqrt{-117})/2)=26\mathcal{X}680^3=(2^6 \cdot 3 \cdot 5 \cdot 1\mathcal{E} \cdot 25)^3}$

These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,

• ${\displaystyle e^{\pi \sqrt{37}} \approx x^{20}-20, x^3-2x^2-2=0}$
• ${\displaystyle e^{\pi \sqrt{57}} \approx x^{20}-20, x^3-2x^2-2x-2=0}$
• ${\displaystyle e^{\pi \sqrt{117}}\approx x^{20}-20, x^3-6x^2+4x-2=0}$

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 20th root, and which explains the 20 in the approximation. In addition, they can also be closely approximated by algebraic numbers of degree 4,

{\displaystyle \begin{align} e^{\pi \sqrt{37}} &\approx 3^5 \left(9-\sqrt{2(1- 680/20+7\sqrt{3\cdot37})} \right)^{-2}-10\\ e^{\pi \sqrt{57}} &\approx 3^5 \left(19-\sqrt{2(1- 3080/20+27\sqrt{3\cdot57})} \right)^{-2}-10\\ e^{\pi \sqrt{117}} &\approx 3^5 \left(173-\sqrt{2(1- 26\mathcal{X}680/20+1491\sqrt{3\cdot117})} \right)^{-2}-10 \end{align} }

If ${\displaystyle x}$ denotes the expression in the parenthesis (e.g. ${\displaystyle x=3-\sqrt{2(1-680/20+1\sqrt{3\cdot37})}}$), it satisfies respectively the quartic equations

{\displaystyle \begin{align} &x^4 - 4\cdot 9x^3 + 0.8(680+3) x^2\ \ \quad\quad- 0.8\cdot 9(680-6)x - 3=0\\ &x^4 - 4\cdot 19x^3 + 0.8(3080+3) x^2\quad\ \;- 0.8\cdot 19(3080-6)x - 3=0\\ &x^4 - 4\cdot 173x^3 + 0.8(26\mathcal{X}680+3) x^2 - 0.8\cdot 173(26\mathcal{X}680-6)x - 3=0\\ \end{align} }

Note the reappearance of the integers ${\displaystyle n = 9, 19, 173}$ as well as the fact that

{\displaystyle \begin{align} &2^6 \cdot 3(-(1- 680/20)^2+7^2\cdot3 \cdot 37) = 680^2\\ &2^6 \cdot 3(-(1- 3080/20)^2+27^2 \cdot 3\cdot 57) = 3080^2\\ &2^6 \cdot 3(-(1- 26\mathcal{X}680/20)^2+1491^2\cdot 3 \cdot 117) = 26\mathcal{X}680^2 \end{align} }

which, with the appropriate fractional power, are precisely the j-invariants.

Also the numbers

• ${\displaystyle e^\pi-\pi=17.\mathcal{E}\mathcal{E}\mathcal{X}54067637\mathcal{E}\mathcal{X}1852\mathcal{X}59537\mathcal{X}\ldots}$
• ${\displaystyle \pi^2-\frac{e}{\pi}=9.007620398766026792165267\ldots}$
• ${\displaystyle \pi^2+\frac{\pi}{20}=\mathcal{X}.000\mathcal{X}552775049609745229\mathcal{X}9\ldots}$

(rounded to 20 significant figures after the dozenal point)

## The first 100 digits after the dozenal point of Ramanujan's constant (${\displaystyle e^{\pi\sqrt{117}}}$)

15059 39952036851E.EEEEEEEEEEE5 391X2899X944 51915EE71E1X 56796X89E046 3508317450E9 6E059X723E15 421185X36493 005802325230 X412E40X6EE4 0E05000X46X6 5E354420E217 60038277590E ...

## Another mathematical coincidence

${\displaystyle \int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)\mathrm{d}x}$ = 0.486701211EE7 8699684E9931 8EX818218908 4X0E2E705496 E9X1732186E4 79924967865X ...

${\displaystyle \frac{\pi}{8}}$ = 0.486701211EE7 8699684E9931 8EX818218908 4X1X173E6129 719E03201391 381984309929 ...

The first 32 digits after the dozenal point of these two values are the same.