# .

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:

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

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

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

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

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

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

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

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

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

another interesting example can be define as the largest root of $${}_{x^7-21 x^5-21 x^4+91 x^3+112 x^2-84 x-97=0} \,,$$ approximate $${}_{4.4934458950490069087} \,$$ and the first positive root of $${}_{\tan\theta=\theta} \,,$$ approximate $${}_{4.49340945790906} \,.$$ What is more, however, the septic is solvable,

$${}_{{}_{\frac{\sqrt[7]{23328}}{6}\left\{\left[-\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}}{12}+ {\rm{i}} \left(-\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2+\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+ \left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+{\rm{i}}\left(\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2+\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+ \left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+{\rm{i}}\left(\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+ \left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+ {\rm{i}} \left(-\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+ \sqrt[7]{-2+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+ \sqrt[7]{-2+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}\right\}}}\,$$

$${}_{\theta} \$$, can also be express in terms of elementary integrals:

$${}_{\theta=\pi e^{\frac{1}{\pi}\int_0^1 \arctan\frac{4\pi t-2\pi t^2 \ln\frac{1+t}{1-t}}{3\pi^2t^2+t^2 \ln^2\frac{1+t}{1-t}-4t \ln\frac{1+t}{1-t}+4}\cdot\frac{\rm{d}t}{t}}=\pi e^{\frac{1}{\pi}\int_0^1 \arctan\frac{4\pi t-4\pi t^2 {\rm{artanh}} t}{3\pi^2t^2+4t^2 {\rm{artanh}} t-8t {\rm{artanh}} t+4}\cdot\frac{\rm{d}t}{t}} } \,$$

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

References

^ a b Eric Weisstein, "Almost Integer" at MathWorld