Almost integer

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%2B\sqrt5}{2}\approx 1.618\,, for example:

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:

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%2B744-2.225\cdots\times 10^{-4}\,
e^{\pi\sqrt{67}}=12^3(21^2-1)^3%2B744-1.337\cdots\times 10^{-6}\,
e^{\pi\sqrt{163}}=12^3(231^2-1)^3%2B744-7.499\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%2B20\,,[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%2B\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%2B20\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%2B\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%2B\sqrt5\right)\Gamma\left({3\over4}\right)}{e^{\frac{5}{6}\pi}}\approx1.000000000000045422} {}_{{1\over4}\left(\cos{1\over10}%2B\cosh{1\over10}%2B2\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%2B\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%2B1\right)%2B1\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}%2B\frac{163}{\ln 163}\approx 4941.99999995925082 }
{}_{\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}%2B4\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}%2B4\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} %2Be^{\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)%2B\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)%2B\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%2B151290x^4-18608670x^3%2B1144433205x^2-28153057165x%2B39605=0} \,
{}_{\frac{615-55\sqrt5-\sqrt[3]{7451370%2B3332354\sqrt5%2B6\sqrt{8890710030%2B3976046490\sqrt5}}-\sqrt[3]{7451370%2B3332354\sqrt5-6\sqrt{8890710030%2B3976046490\sqrt5}}}{6}\approx 1.40677447684\times10^{-6}}
{}_{\qquad\mbox{Root of } 312500000x^5-6843750000x^4%2B6826250000x^3%2B10476025000x^2-7886869750x-72099=0} \,
{}_{\tan\left(\frac{\arctan 4}{5}%2B\frac{4\pi}{5}\right)%2B\frac{19}{50}=\frac{219}{50}%2B\frac{-1-\sqrt5%2B\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884%2B799{\rm{i}}}%2B\frac{-1-\sqrt5-\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884-799{\rm{i}}}%2B\frac{-1%2B\sqrt5-\sqrt{10%2B2\sqrt5}{\rm{i}}}{4}\sqrt[5]{1156%2B289{\rm{i}}}%2B\frac{-1%2B\sqrt5%2B\sqrt{10%2B2\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%2Bx}\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%2Bx}\ln \left[\Gamma\left(2-x\right)\right]\ln \left[\Gamma\left(2%2Bx\right)\right]{\rm{d}} x}=\sqrt2e^{\frac{\pi^2}{12\ln 2}%2B\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 lagerst root of {}_{x^7-21 x^5-21 x^4%2B91 x^3%2B112 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}%2B\frac{\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\sqrt[3]{28-84\sqrt3{\rm{i}}}}{12}%2B {\rm{i}} \left(-\frac{\sqrt7}{6}%2B\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7%2B12\sqrt{21}{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2%2B\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(-14%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548%2B588\sqrt3{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}%2B \left[ -\frac{1}{6}%2B\frac{\sqrt[3]{28%2B84\sqrt{3}{\rm{i}}}%2B\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}%2B{\rm{i}}\left(\frac{\sqrt7}{6}%2B\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7%2B12\sqrt{21}{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2%2B\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(14%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548%2B588\sqrt3{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}%2B \left[ -\frac{1}{6}%2B\frac{\sqrt[3]{28%2B84\sqrt{3}{\rm{i}}}%2B\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}%2B{\rm{i}}\left(\frac{\sqrt7}{6}%2B\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7%2B12\sqrt{21}{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(-14%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{2548%2B588\sqrt3{\rm{i}}}%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}%2B \left[ -\frac{1}{6}%2B\frac{\sqrt[3]{28%2B84\sqrt{3}{\rm{i}}}%2B\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}%2B {\rm{i}} \left(-\frac{\sqrt7}{6}%2B\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7%2B12\sqrt{21}{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right] \sqrt[7]{-2%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(14%2B\sqrt[3]{-2548%2B588\sqrt3{\rm{i}}}%2B\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}%2B \sqrt[7]{-2%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(-14%2B\sqrt[3]{2548%2B588\sqrt3{\rm{i}}}%2B\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}%2B \sqrt[7]{-2%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{28%2B84\sqrt3{\rm{i}}}%2B\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}%2B\frac{\sqrt7}{7}\left(14%2B\frac{-1%2B\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548%2B588\sqrt3{\rm{i}}}%2B\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%2Bt}{1-t}}{3\pi^2t^2%2Bt^2 \ln^2\frac{1%2Bt}{1-t}-4t \ln\frac{1%2Bt}{1-t}%2B4}\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%2B4t^2 {\rm{artanh}} t-8t {\rm{artanh}} t%2B4}\cdot\frac{\rm{d}t}{t}} } \,

External links

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#