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.