Almost integer explained

In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected.

Almost integers relating to the golden ratio and Fibonacci numbers

\phi = (1+ \sqrt 5)/(2) ≈ 1 . 618 , for example:\begin{align} \phi ^{17} & = (3571+1597 \sqrt 5)/(2) ≈ 3571 . 00028 \\[ 6 pt] \phi ^{18} & =2889+1292 \sqrt 5 ≈ 5777 . 999827 \\[ 6 pt] \phi ^{19} & = (9349+4181 \sqrt 5)/(2) ≈ 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:

(\operatorname{Fib)/(( 360 )}{\operatorname{Fib}( 216 )}) ≈ 1242282009792667284144565908481 . 999999999999999999999999999999195 (\operatorname{Lucas)/(( 361 )}{\operatorname{Lucas}( 216 )}) ≈ 2010054515457065378082322433761 . 000000000000000000000000000000497

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

a(n) = (\operatorname{Fib)/(( 45 x 2) ^{n)}{\operatorname{Fib}(27 x 2} ^{n)}} ≈ \operatorname{Lucas}( 18 x 2 ^{n)} a(n) = (\operatorname{Lucas)/(( 45 x 2) ^{n+1)}{\operatorname{Lucas}(27 x 2} ^{n)}} ≈ \operatorname{Lucas}( 18 x 2 ^{n+1)} As n increases, the number of consecutive nines or zeros beginning at the tenths place of a(n) approaches infinity.

Almost integers relating to e and

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

e ^{\pi\sqrt{43} }\approx 884736743.999777466e ^{\pi\sqrt{67} }\approx 147197952743.999998662454e ^{\pi\sqrt{163} }\approx 262537412640768743.99999999999925007where the non-coincidence can be better appreciated when expressed in the common simple form:^[1] e ^{\pi\sqrt{43} }=12^3(9^2-1)^3+744-(2.225\ldots)\times 10^e ^{\pi\sqrt{67} }=12^3(21^2-1)^3+744-(1.337\ldots)\times 10^e ^{\pi\sqrt{163} }=12^3(231^2-1)^3+744-(7.499\ldots)\times 10^where21=3 x 7 , 231=3 x 7 x 11 , 744=24 x 31 and the reason for the squares is 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 and e have often puzzled mathematicians. An example is:

e ^{\pi-\pi=19.999099979189\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:$$\backslash sum\_^\backslash left(8\backslash pi\; k^2\; -2\; \backslash right)\; e^\; =\; 1.$$The first term dominates since the sum of the terms for k\geq 2 total \sim 0 . 0003436 . The sum can therefore be truncated to \left( 8 \pi -2 \right) e ^{-\pi} ≈ 1 , where solving for e ^{\pi} gives e ^{\pi} ≈ 8 \pi -2 . Rewriting the approximation for e ^{\pi} and using the approximation for 7 \pi ≈ 22 gives $$e^\; \backslash approx\; \backslash pi\; +\; 7\backslash pi\; -\; 2\; \backslash approx\; \backslash pi\; +\; 22-2\; =\; \backslash pi+20.$$Thus, rearranging terms gives e ^{\pi} - \pi ≈ 20 . Ironically, the crude approximation for 7 \pi yields an additional order of magnitude of precision.^[2]

Another example involving these constants is:

e + \pi + e\pi + e ^{\pi+\pi} ^{e=59.9994590558\ldots}

See also

• Schizophrenic number

External links

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

Notes and References

1. Web site: More on e^(pi*SQRT(163)).
2. [Eric Weisstein]