almost integer

Some examples of almost integers are high powers of the golden ratio $\backslash phi=\backslash frac\{1+\backslash sqrt5\}\{2\}\backslash approx\; 1.618$, for example:

: $$

\begin{align}

\phi^{17} & =\frac{3571+1597\sqrt5}{2}\approx 3571.00028 \\[6pt]

\phi^{18} & =2889+1292\sqrt5 \approx 5777.999827 \\[6pt]

\phi^{19} & =\frac{9349+4181\sqrt5}{2}\approx 9349.000107

\end{align}

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

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

• $\backslash frac\{\backslash operatorname\{Fib\}(360)\}\{\backslash operatorname\{Fib\}(216)\}\; \backslash approx\; 1242282009792667284144565908481.999999999999999999999999999999195$
• $\backslash frac\{\backslash operatorname\{Lucas\}(361)\}\{\backslash operatorname\{Lucas\}(216)\}\; \backslash approx\; 2010054515457065378082322433761.000000000000000000000000000000497$

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

• $a(n)\; =\; \backslash frac\{\backslash operatorname\{Fib\}(45\backslash times2^n)\}\{\backslash operatorname\{Fib\}(27\backslash times2^n)\}\; \backslash approx\; \backslash operatorname\{Lucas\}(18\backslash times2^n)$
• $a(n)\; =\; \backslash frac\{\backslash operatorname\{Lucas\}(45\backslash times2^n+1)\}\{\backslash operatorname\{Lucas\}(27\backslash times2^n)\}\; \backslash approx\; \backslash operatorname\{Lucas\}(18\backslash times2^n+1)$

As n increases, the number of consecutive nines or zeros beginning at the tenths place of a(n) approaches infinity.

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

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

where the non-coincidence can be better appreciated when expressed in the common simple form:{{Cite web|url=http://groups.google.com/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#|title=More on e^(pi*SQRT(163))}}

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

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

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

where

:$21=3\backslash times7,\; \backslash quad\; 231=3\backslash times7\backslash times11,\; \backslash quad\; 744=24\backslash times\; 31$

and the reason for the squares is due to certain Eisenstein series. The constant $e^\{\backslash pi\backslash sqrt\{163\}\}$

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: $e^\backslash pi-\backslash pi=19.999099979189\backslash ldots$

The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows:

$$\backslash sum\_\{k=1\}^\{\backslash infty\}\backslash left(\; 8\backslash pi\; k^2\; -2\; \backslash right)\; e^\{-\backslash pi\; k^2\}\; =\; 1.$$

The first term dominates since the sum of the terms for $k\backslash geq\; 2$ total $\backslash sim\; 0.0003436.$ The sum can therefore be truncated to

$\backslash left(\; 8\backslash pi\; -2\backslash right)\; e^\{-\backslash pi\}\backslash approx\; 1,$

where solving for $e^\{\backslash pi\}$ gives $e^\{\backslash pi\}\; \backslash approx\; 8\backslash pi\; -2.$

Rewriting the approximation for $e^\{\backslash pi\}$ and using the approximation for $7\backslash pi\; \backslash approx\; 22$ gives

$$e^\{\backslash pi\}\; \backslash approx\; \backslash pi\; +\; 7\backslash pi\; -\; 2\; \backslash approx\; \backslash pi\; +\; 22-2\; =\; \backslash pi+20.$$

Thus, rearranging terms gives $e^\{\backslash pi\}\; -\; \backslash pi\; \backslash approx\; 20.$ Ironically, the crude approximation for $7\backslash pi$ yields an additional order of magnitude of precision.

Eric Weisstein, [http://mathworld.wolfram.com/AlmostInteger.html "Almost Integer"] at MathWorld

Another example involving these constants is: $e+\backslash pi+e\backslash pi+e^\backslash pi+\backslash pi^e=59.9994590558\backslash ldots$

• Schizophrenic number

{{Reflist}}

• [http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought]

Category:Integers

Category:Recreational mathematics