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

Almost integers related to e and π[]

$e^{\pi {\sqrt {37}}}=20836851{\mathcal {E}}.{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}7476257520{\mathcal {E}}483574{\mathcal {X}}86{\mathcal {X}}\ldots$

$e^{\pi {\sqrt {57}}}=2464036851{\mathcal {E}}.{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}800{\mathcal {X}}683617{\mathcal {E}}91255{\mathcal {E}}09\ldots$

$e^{\pi {\sqrt {117}}}=1505939952036851{\mathcal {E}}.{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}{\mathcal {E}}5391{\mathcal {X}}2899{\mathcal {X}}944\ldots$

(rounded to 20 significant figures after the dozenal point)

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

The forms of them are

$e^{\pi {\sqrt {37}}}=680^{3}+520-4.74596469{\mathcal {E}}07\ldots \times 10^{-4}$

$e^{\pi {\sqrt {57}}}=3080^{3}+520-3.{\mathcal {E}}{\mathcal {E}}15385{\mathcal {X}}402\ldots \times 10^{-6}$

$e^{\pi {\sqrt {117}}}=26{\mathcal {X}}680^{3}+520-6.82{\mathcal {X}}19322127\ldots \times 10^{-10}$

Alternatively,

$e^{\pi {\sqrt {37}}}=10^{3}(9^{2}-1)^{3}+520-4.74596469{\mathcal {E}}07\ldots \times 10^{-4}$

$e^{\pi {\sqrt {57}}}=10^{3}(19^{2}-1)^{3}+520-3.{\mathcal {E}}{\mathcal {E}}15385{\mathcal {X}}402\ldots \times 10^{-6}$

$e^{\pi {\sqrt {117}}}=10^{3}(173^{2}-1)^{3}+520-6.82{\mathcal {X}}19322127\ldots \times 10^{-10}$

(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 $d<17$ , one does not obtain an almost integer; even $d=17$ is not noteworthy (because of d = 17, the form is $e^{\pi {\sqrt {17}}}=80^{3}+520-0.280203603728\ldots$ , but the absolute deviation of a random real number (picked uniformly from $[0,1]$ , say) is a uniformly distributed variable on $[0,0.6]$ , 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 $10^{3}(n^{2}-1)^{3}=(2^{2}\cdot 3\cdot (n-1)\cdot (n+1))^{3}$ form, and factor as,

$j((1+{\sqrt {-37}})/2)=680^{3}=(2^{6}\cdot 3\cdot 5)^{3}$

$j((1+{\sqrt {-57}})/2)=3080^{3}=(2^{5}\cdot 3\cdot 5\cdot {\mathcal {E}})^{3}$

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

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,

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

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,

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

If $x$ denotes the expression in the parenthesis (e.g. $x=3-{\sqrt {2(1-680/20+1{\sqrt {3\cdot 37}})}}$ ), it satisfies respectively the quartic equations

{\begin{aligned}&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{aligned}}

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

{\begin{aligned}&2^{6}\cdot 3(-(1-680/20)^{2}+7^{2}\cdot 3\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{aligned}}

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

Also the numbers

$e^{\pi }-\pi =17.{\mathcal {E}}{\mathcal {E}}{\mathcal {X}}54067637{\mathcal {E}}{\mathcal {X}}1852{\mathcal {X}}59537{\mathcal {X}}\ldots$

$\pi ^{2}-{\frac {e}{\pi }}=9.007620398766026792165267\ldots$

$\pi ^{2}+{\frac {\pi }{20}}={\mathcal {X}}.000{\mathcal {X}}552775049609745229{\mathcal {X}}9\ldots$

(rounded to 20 significant figures after the dozenal point)

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

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

Another mathematical coincidence[]

$\int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos \left({\frac {x}{n}}\right)\mathrm {d} x$ = 0.486701211EE7 8699684E9931 8EX818218908 4X0E2E705496 E9X1732186E4 79924967865X ...

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

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

Almost integer

Almost integers related to e and π[]

The first 100 digits after the dozenal point of Ramanujan's constant ()[]

Another mathematical coincidence[]

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