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.

Almost Integers relating to the Golden Ratio and Fibonacci Numbers

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.

The ratios of Fibonacci and Lucas numbers can also make countless almost integers, for instance:

$Fib(360)/Fib(216)\approx 1242282009792667284144565908481.999999999999999999999999999999195$
$Lucas(361)/Lucas(216)\approx 2010054515457065378082322433761.000000000000000000000000000000497$

The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:

$a(n)=Fib(45*2^n)/Fib(27*2^n)\approx Lucas(18*2^n)$
$a(n)=Lucas(45*2^n+1)/Lucas(27*2^n)\approx Lucas(18*2^n+1)$

As n increases, the consecutive nines (or zeros) at the tenth place of a(n) approach infinity, as illustrated respectively by (sequence A268665 in OEIS) and (sequence A270073 in OEIS).

Almost Integers relating to e and π

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 that involve 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 considered a mathematical coincidence.

References

External links

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

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Almost integer

Almost Integers relating to the Golden Ratio and Fibonacci Numbers

Almost Integers relating to e and π

See also

References

External links