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The fact that these powers approach integers is non-coincidental, which is trivially seen because the golden ratio is a Pisot-Vijayaraghavan number.
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:
Almost integers that involve the mathematical constants pi 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 , which is therefore considered a mathematical coincidence.