A noble number
is defined as an irrational number having a
continued fraction that becomes an infinite
sequence of 1s at some point,
The prototype is the inverse of the golden ratio , whose continued
fraction is composed entirely of 1s (except for the term), .
Any noble number can be written as
and are the numerator
and denominator of the th convergent of .
The noble numbers are a subset of but not a subfield,
since there is no subfield lying properly between and . To see this, consider , which must be contained in the same field as
but is not a noble number since its
continued fraction is .
More things to try:
Weisstein, Eric W. "Noble Number." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NobleNumber.html