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Fundamental Unit

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For a squarefree integer , the number is a fundamental unit if . Basically, there are no fractions in either the number or its reciprocal.

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For example, the fundamental unit of is . The reciprocal of the fundamental unit is .

For squarefree integers (mod 4), the value is a fundamental unit if . In these cases, is allowed.

Thus, the fundamental unit is either or . Except for the golden ratio, one of the continued fraction convergents of is .

Also, equals one of the values . There are no smaller integers and that give that value for the given .

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Contributed by: Ed Pegg Jr (June 13)
Open content licensed under CC BY-NC-SA

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