Almost integer
In recreational mathematics, an almost integer is any number that is very close to an integer. Almost integers are considered interesting when they arise in some context in which they are unexpected.
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Almost integers relating to the golden ratio and Fibonacci numbers
Well-known examples of almost integers are high powers of the golden ratio , 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.
The ratios of Fibonacci or Lucas numbers can also make countless almost integers, for instance:
The above examples can be generalized by the following sequences, which generate near-integers approaching Lucas numbers with increasing precision:
As n increases, the number of consecutive nines or zeros beginning at the tenth place of a(n) approaches infinity.
Almost integers relating to e and π
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]}
where
and the reason for the squares being due to certain Eisenstein series. The constant 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: To date, no explanation has been given for why Gelfond's constant () is nearly identical to ,^{[1]} which is therefore considered a mathematical coincidence.
See also
References
- ^ ^{a} ^{b} Eric Weisstein, "Almost Integer" at MathWorld
- ^ http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#
External links
- J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought