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

*n*th silver mean is given by the formula (

*n* + sqrt(

*n*^{2} + 4))/2.

The *n*th 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.