silver mean (idea)

A positive number is a silver mean if it differs from its reciprocal by an integer. The name is a generalisation of the term golden mean, which is used to refer to the unique positive number which is 1 more than its reciprocal. (Various powers of the golden ratio are silver means.) The nth silver mean is given by the formula (n + sqrt(n² + 4))/2.

The nth silver mean has a very simple continued fraction representation in terms of n. For example, consider this continued fraction:

2 + 1
   _______
    2 + 1
       _______
        2 + 1
           _______
            2 + 1
               _______
                2 + ...

It should be obvious that subtracting two from this number and then taking the reciprocal leaves the whole expression unchanged; in other words, this number is two more than its reciprocal and hence is the second silver mean.