Almost integer
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 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, because the golden ratio is a Pisot–Vijayaraghavan number.
The ratios of Fibonacci or Lucas numbers can also make 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 tenths 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 is 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.