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TRANSCEHDESITAL HUMBEKS
BASED OH THE FIBONACCI SEQUENCE
DONALD KNUTH
California Institute of Technology, Pasadena, California
A well-known theorem due to Liouville states that if £ is an irrational
algebraic number of degree n, then the equation
S-
(1)
R
q
n+e
has only finitely many solutions for integers p,q, given any e > 0.
There-
fore, an irrational number £, for which
(2)
q
has solutions for arbitrarily large t, must be transcendental.
Numbers of
this type have been called Liouville numbers.
In 1955, Roth published his celebrated improvement of Liouville f s theorem,
replacing "nM by "2" in equation (1). Let us call an irrational number £,
for which
(3)
q
2+e
his infinitely many solutions for some € > 0, a Roth number.
Roth numbers
are also transcendental, and they include many more numbers than the Liouville
numbers.
Let b be an integer greater than 1. Then we define ^,
to be the con-
tinued fraction
(4)
,b
""'' + ?r-£ +
Theorem: £, is a Roth number, hence i|,
43
is transcendental.
44
TRANSCENDENTAL NUMBERS BASED ON
Proof:
F r o m the e l e m e n t a r y t h e o r y of continued fractions, it is well
known that if p / q
is the n
*n m
convergent
to •£,,$
fe
h
P
(5)
n
q 0 = 1,
then
-i /
—q
Fm i
In this c a s e ,
[Feb. 1964]
q t = b °f
< i / q q . -i
*n Ti-i-1
^n
and q
= b
' q
+ q
. We can t h e r e -
fore e a s i l y verify by induction that
K
(6)
q.n
Fn+i _
i
b - 1
Fn+2/ Fn+i
Fn
In p a r t i c u l a r , as n -*• <*> we have q , - , / q
-* b
/ b
= b
. ~ [(b l ) q y w h e r e <p = .618 • • • i s the golden r a t i o . T h e r e f o r e for l a r g e n we
f
have approximately
2+cf)
n
and this c o m p l e t e s the proof of the t h e o r e m ,
R e m a r k s , It can be easily shown that the s e t of Roth n u m b e r s i s of m e a s u r e
F o r e x a m p l e , the n u m b e r 2 . b n » w h e r e {c }
F
L
*
n=l
nJ
i s a s t r i c t l y i n c r e a s i n g s e q u e n c e of positive i n t e g e r s , i s a Roth n u m b e r ifl i m sup v(c - / c ;) > 2, and it is a Liouville n u m b e r if this l i m sup i s i n n->oo^
n+1/ n
^
finite. In t e r m s of continued f r a c t i o n s , the n u m b e r
z e r o , but it is uncountable.
J,
i
a1 + ™ , 1
1
a
2
a* +
i s a Roth n u m b e r if and only if
l i m sup (log a i V / log q n )
where q
> 0
a r e the d e n o m i n a t o r s as in the proof of the above t h e o r e m .
The r a p i d c o n v e r g e n c e of (4) allows us to evaluate £,
p r e c i s i o n , e. g„ ,
R e f e r e n c e to this a r t i c l e on p 9
£2 = .70980 34448 61291 • • •
^ 3 = ,76859 75625 93155 • • •
520
easily with high