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Ed Pegg, Jr. noted that the length d equals that is very close to 7 (7.0000000857 ca.)[1]

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 numbersEdit

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 countless almost integers, for instance:

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

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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 πEdit

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

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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.

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