<s>
In	O
number	O
theory	O
,	O
Carmichael	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
named	O
after	O
the	O
American	O
mathematician	O
R	O
.	O
D	O
.	O
Carmichael	O
,	O
</s>
<s>
states	O
that	O
,	O
for	O
any	O
nondegenerate	O
Lucas	B-Algorithm
sequence	I-Algorithm
of	O
the	O
first	O
kind	O
Un(P,Q )	O
with	O
relatively	O
prime	O
parameters	O
P	O
,	O
Q	O
and	O
positive	O
discriminant	O
,	O
an	O
element	O
Un	O
with	O
n≠1	O
,	O
2	O
,	O
6	O
has	O
at	O
least	O
one	O
prime	O
divisor	O
that	O
does	O
not	O
divide	O
any	O
earlier	O
one	O
except	O
the	O
12th	O
Fibonacci	B-Algorithm
number	I-Algorithm
F(12 )	O
=	O
U12( 1	O
,	O
−1	O
)	O
=	O
144	O
and	O
its	O
equivalent	O
U12( −1	O
,	O
−1	O
)	O
=	O
−144	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
n	O
greater	O
than	O
12	O
,	O
the	O
nth	O
Fibonacci	B-Algorithm
number	I-Algorithm
F(n )	O
has	O
at	O
least	O
one	O
prime	O
divisor	O
that	O
does	O
not	O
divide	O
any	O
earlier	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Carmichael	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
says	O
that	O
every	O
Fibonacci	B-Algorithm
number	I-Algorithm
,	O
apart	O
from	O
the	O
exceptions	O
listed	O
above	O
,	O
has	O
at	O
least	O
one	O
primitive	O
prime	O
divisor	O
.	O
</s>
