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In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. Also, every known pair shares at least one common factor, higher than 1. It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.
In 1955, Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.
Read more about this topic: Amicable Numbers
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