Almost integer

Well-known examples of almost integers are high powers of the golden ratio ${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}$, for example:

${\displaystyle {\begin{aligned}\phi ^{17}&={\frac {3571+1597{\sqrt {5}}}{2}}\approx 3571.00028\\[6pt]\phi ^{18}&=2889+1292{\sqrt {5}}\approx 5777.999827\\[6pt]\phi ^{19}&={\frac {9349+4181{\sqrt {5}}}{2}}\approx 9349.000107\end{aligned}}}$

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 countless almost integers, for instance:

• ${\displaystyle \operatorname {Fib} (360)/\operatorname {Fib} (216)\approx 1242282009792667284144565908481.999999999999999999999999999999195}$
• ${\displaystyle \operatorname {Lucas} (361)/\operatorname {Lucas} (216)\approx 2010054515457065378082322433761.000000000000000000000000000000497}$

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

• ${\displaystyle a(n)=\operatorname {Fib} (45\times 2^{n})/\operatorname {Fib} (27\times 2^{n})\approx \operatorname {Lucas} (18\times 2^{n})}$
• ${\displaystyle a(n)=\operatorname {Lucas} (45\times 2^{n}+1)/\operatorname {Lucas} (27\times 2^{n})\approx \operatorname {Lucas} (18\times 2^{n}+1)}$

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

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

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

where the non-coincidence can be better appreciated when expressed in the common simple form:[2]

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

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

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

where

${\displaystyle 21=3\times 7,\quad 231=3\times 7\times 11,\quad 744=24\times 31}$

and the reason for the squares being due to certain Eisenstein series. The constant ${\displaystyle e^{\pi {\sqrt {163}}}}$ is sometimes referred to as Ramanujan's constant.

Almost integers that involve the mathematical constants π and e have often puzzled mathematicians. An example is: ${\displaystyle e^{\pi }-\pi =19.999099979189\ldots }$ To date, no explanation has been given for why Gelfond's constant (${\displaystyle e^{\pi }}$) is nearly identical to ${\displaystyle \pi +20}$,[1] which is therefore considered a mathematical coincidence.

• Schizophrenic number

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