Almost integer

Ed Pegg, Jr. noted that the length d equals $\frac{1}{2}\sqrt{\frac{1}{30}(61421-23\sqrt{5831385})}$ that is very close to 7 (7.0000000857 ca.)^[1]

In recreational mathematics an almost integer is any number that is very close to an integer. Well known examples of almost integers are high powers of the golden ratio $\phi=\frac{1+\sqrt5}{2}\approx 1.618\,$, for example:

• $\phi^{17}=\frac{3571+1597\sqrt5}{2}\approx 3571.00028\,$
• $\phi^{18}=2889+1292\sqrt5 \approx 5777.999827\,$
• $\phi^{19}=\frac{9349+4181\sqrt5}{2}\approx 9349.000107\,$

The fact that these powers approach integers is non-coincidental, which is trivially seen because the golden ratio is a Pisot-Vijayaraghavan number.

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

• $e^{\pi\sqrt{43}}\approx 884736743.999777466\,$
• $e^{\pi\sqrt{67}}\approx 147197952743.999998662454\,$
• $e^{\pi\sqrt{163}}\approx 262537412640768743.99999999999925007\,$

where the non-coincidence can be better appreciated when expressed in the common simple form^[2]:

$e^{\pi\sqrt{43}}=12^3(9^2-1)^3+744-2.225\cdots\times 10^{-4}\,$

$e^{\pi\sqrt{67}}=12^3(21^2-1)^3+744-1.337\cdots\times 10^{-6}\,$

$e^{\pi\sqrt{163}}=12^3(231^2-1)^3+744-7.499\cdots\times 10^{-13}\,$

where :$21=3\times7,231=3\times7\times11,744=24\times 31\,$ and the reason for the squares being due to certain Eisenstein series. The constant $e^{\pi\sqrt{163}}\,$ is sometimes referred to as Ramanujan's constant.

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

$e^{\pi}-\pi=19.999099979189\cdots\,$

To date, no explanation has been given for why Gelfond's constant ( $e^{\pi}\,$ ) is nearly identical to $\pi+20\,$,^[1] which is therefore regarded to be a mathematical coincidence.

Another example is

$22{\pi}^4=2143.0000027480\cdots\,$

Also consider π in cubic expressions

${\pi}^3=31.006276\cdots\,$

or

${\pi}^3-\frac{\pi}{500}=30.999993494\cdots\,$

where the second one is obvious from the first one.

Also consider π in quadratic expressions

${\pi}^2= 9.8696044\cdots\,$

or

${\pi}^2+\frac{\pi}{24}=10.000504\cdots\,$

where the second one is obvious from the first one.

Here are more examples:

${}_{\cos\left\{\pi\cos\left[\pi\cos\ln\left(\pi+20\right)\right]\right\}\approx -0.9999999999999999999999999999999999606783 }$	${}_{\sin2017\sqrt[5]2\approx -0.9999999999999999785}$	${}_{\sum_{k=1}^{\infty}\frac{\lfloor n\tanh \pi \rfloor}{10^n}-\frac{1}{81}\approx 1.11\times10^{-269}}$	${}_{\sqrt{29}\left(\cos\frac{2\pi}{59}-\cos\frac{24\pi}{59}\right)-\frac{19}{5}\approx 3.057684294154\times10^{-6}}$
${}_{1+\frac{103378831900730205293632}{e^{3\pi\sqrt{163}}}-\frac{196884}{e^{2\pi\sqrt{163}}}-\frac{262537412640768744}{e^{\pi\sqrt{163}}}\approx 1.161367900476\times10^{-59}}$	${}_{\frac{\ln^2262537412640768744}{\pi^2}-163\approx 2.32167\times10^{-29}}$	${}_{10\tanh\frac{28}{15}\pi-\frac{\pi^9}{e^8}\approx 3.661398\times10^{-8}}$	${}_{ \sqrt[4]{\frac{91}{10}}-\frac{33}{19}\approx 3.661398\times10^{-8}}$
${}_{ \gamma-{10\over81}\left(11-2\sqrt{10}\right)=\int_0^{\infty}\left(\frac{1}{e^x-1}-\frac{1}{xe^x}\right){\rm{d}}x-{10\over81}\left(11-2\sqrt{10}\right)\approx 2.72\times10^{-7}}$	${}_{\frac{\left(5+\sqrt5\right)\Gamma\left({3\over4}\right)}{e^{\frac{5}{6}\pi}}\approx1.000000000000045422}$	${}_{{1\over4}\left(\cos{1\over10}+\cosh{1\over10}+2\cos{\sqrt2\over20}\cosh{\sqrt2\over20}\right)\approx 1.000000000000248 }$	${}_{e^6-\pi^5-\pi^4\approx1.7673\times10^{-5}}$
${}_{\sqrt{29}\left(\cos\frac{2\pi}{59}-\cos\frac{24\pi}{59}\right)\approx 3.0576842941540143382\times 10^{-6}}$	${}_{ \ln K-\ln\ln K\approx 1.0000744}$	${}_{ e^{\phi_0\left(\frac{2+\sqrt3}{4}\right)}=e^{\int_0^{\infty}\left(\frac{1}{te^t}-\frac{e^{\frac{2-\sqrt3}{4}t}}{e^t-1}\right){\rm{d}}t}\approx 1.99999969}$	${}_{ \frac{\sqrt[3]9}{3\ln 2}\approx 1.00030887}$
${}_{\sum_{k=-\infty}^{\infty}10^{-\frac{k^2}{10000}}-100\sqrt{\frac{\pi}{\ln10}}=\theta_3\left(0,\frac{1}{\sqrt[10000]{10}}\right)-100\sqrt{\frac{\pi}{\ln10}}\approx1.3809\times10^{-18613}}$	${}_{ {\pi^9\over e^8}\approx 9.998387}$	${}_{ e^{\pi}-\pi\approx 19.999099979}$	${}_{ \frac{e^{\pi}-\ln3}{\ln2}-\frac{4}{5}\approx 31.0000000033}$
${}_{\frac{\pi^{11}}{e^3}-\Gamma\left[\Gamma\left(\pi+1\right)+1\right]=\frac{\pi^{11}}{e^3}-\int_0^{\infty}\frac{t^{\int_0^{\infty}\frac{u^{\pi}}{e^u}{\rm{d}}u}}{e^t} {\rm{d}}t\approx 7266.9999993632596}$	${}_{ 163\left(\pi-e\right)\approx 68.999664}$	${}_{ \left(\frac{23}{9}\right)^5=\frac{6436343}{59049}\approx 109.00003387}$	${}_{ 88\ln 89\approx 395.00000053}$
${}_{ 510\lg 7\approx 431.00000040727098}$	${}_{ 272\log_{\pi}97\approx 1087.000000204}$	${}_{ \frac{53453}{\ln 53453}\approx 4910.00000122}$	${}_{ \frac{53453}{\ln 53453}+\frac{163}{\ln 163}\approx 4941.99999995925082 }$
${}_{\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}+4\pi e^{\pi}\right)}-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}-4\pi e^{\pi}\right)}\approx 2.570287024592328869357\times 10^{-6}}$	${}_{10-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}-4\pi e^{\pi}\right)}\approx 2.57055302118\times 10^{-6}}$	${}_{10-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}+4\pi e^{\pi}\right)}\approx 2.65996596963\times 10^{-10}}$	${}_{ \frac{163}{\ln 163}\approx 31.9999987343}$
${}_{ \left(3\sqrt5\right)^{\gamma}=\left(3\sqrt5\right)^{\int_0^{\infty}\left(\frac{1}{e^x-1}-\frac{1}{xe^x}\right){\rm{d}}x}\approx 3.000060964}$

${}_{\frac{10}{81}-\sum_{n=1}^\infty\frac{\sum_{k=10^{n-1}}^{10^n-1}10^{-n\left[k-(10^{n-1}-1)\right]}k}{10^{\sum_{k=0}^{n-1}9\times 10^{k-1}k}}=\frac{10}{81}-\sum_{n=1}^\infty\sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{kn-9\sum_{k=0}^{n-1}10^k(n-k)}}\approx 1.022344\times10^{-9}}$

${}_{-\frac{1}{5} +e^{\frac{6}{5}} {}_4F_3\left(-\frac{1}{5},\frac{1}{20},\frac{3}{10},\frac{11}{20};\frac{1}{5},\frac{2}{5},\frac{3}{5};\frac{256}{3125e^6}\right)+\frac{2}{25e^{\frac{6}{5}}}{}_4F_3\left(\frac{1}{5},\frac{9}{20},\frac{7}{10},\frac{19}{20};\frac{3}{5},\frac{4}{5},\frac{7}{5};\frac{256}{3125e^6}\right)-\frac{4}{125e^{\frac{12}{5}}}{}_4F_3\left(\frac{2}{5},\frac{13}{20},\frac{9}{10},\frac{23}{20};\frac{4}{5},\frac{6}{5},\frac{8}{5};\frac{256}{3125e^6}\right)+\frac{7}{625e^{\frac{18}{5}}}{}_4F_3\left(\frac{3}{5},\frac{17}{20},\frac{11}{10},\frac{27}{20};\frac{6}{5},\frac{7}{5},\frac{9}{5};\frac{256}{3125e^6}\right)-\pi\approx 2.89221114964408683\times10^{-8}}$

${}_{\qquad\mbox{Root of } x^6-615x^5+151290x^4-18608670x^3+1144433205x^2-28153057165x+39605=0} \,$

${}_{\frac{615-55\sqrt5-\sqrt[3]{7451370+3332354\sqrt5+6\sqrt{8890710030+3976046490\sqrt5}}-\sqrt[3]{7451370+3332354\sqrt5-6\sqrt{8890710030+3976046490\sqrt5}}}{6}\approx 1.40677447684\times10^{-6}}$

${}_{\qquad\mbox{Root of } 312500000x^5-6843750000x^4+6826250000x^3+10476025000x^2-7886869750x-72099=0} \,$

${}_{\tan\left(\frac{\arctan 4}{5}+\frac{4\pi}{5}\right)+\frac{19}{50}=\frac{219}{50}+\frac{-1-\sqrt5+\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884+799{\rm{i}}}+\frac{-1-\sqrt5-\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884-799{\rm{i}}}+\frac{-1+\sqrt5-\sqrt{10+2\sqrt5}{\rm{i}}}{4}\sqrt[5]{1156+289{\rm{i}}}+\frac{-1+\sqrt5+\sqrt{10+2\sqrt5}{\rm{i}}}{4}\sqrt[5]{1156-289{\rm{i}}}\approx -9.141538637378949398666277\times 10^{-6}}$

${}_{\rm{erfi}\left(\rm{erfi}\frac{\sqrt3}{3}\right)=\frac{2}{\sqrt\pi}\int_0^{\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t} e^{u^2} \rm{d} u =\frac{2}{\sqrt\pi}e^{\left(\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right)^2}\int_0^{\infty}\frac{\sin\left[\frac{4u\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right]}{e^{u^2}}{\rm{d}}u =\frac{2}{\sqrt\pi}\int_0^{{}_{\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t}} e^{u^2} {\rm{d}} u =\frac{2}{\sqrt\pi}e^{\left(\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)^2}\int_0^{\infty}\frac{\sin\left(\frac{4u}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)}{e^{u^2}} {\rm{d}} u\approx 1.00002087363809430195879}$

${}_{K=2e^{\frac{1}{\ln 2}\int_0^1\frac{1}{x^2+x}\ln \frac{\pi\left(x-x^3\right)}{\sin\left(\pi x\right)}{\rm{d}} x}=2e^{\frac{1}{\ln 2}\int_0^1\frac{1}{x^2+x}\ln \left[\Gamma\left(2-x\right)\right]\ln \left[\Gamma\left(2+x\right)\right]{\rm{d}} x}=\sqrt2e^{\frac{\pi^2}{12\ln 2}+\frac{1}{\ln 2}\int_0^{\pi}\frac{\ln\left(\theta|\cot\theta|\right)}{\theta}{\rm{d}} \theta}}$

External links

• J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought

References

1. ^ ^{a b} Eric Weisstein, "Almost Integer" at MathWorld
2. ^ http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#