16
GENERALIZATIONS OF THE PERRON-FROBENIUS THEOREM
such that fovjkfi equation (3.22) holds. Since
k k
1 y^fi
+
Sfc+1
_ y^ H-ih ~ a ,
sfc+i
k k
3 = 1
J
3 = \
J
^
=
1 _ It. +
g/c+1
Tk rk+i'
it follows that Sk+i lk+ilk So we can choose 7^+1 0 such that s/e+i =
7 ^ + 1 7^+1 and this proves (3.22). Recall also that for j = 1, 2,... , //
r^s for every s Sj.
With this information we can use rule B to prove that lcm(S') P(n).
Define integers q^ 1 i /i, by
o7 = 1cm (lcm(5M _i+i),lcm(SM_i+2),... ,lcm(5A,)J.
We shall use rule B to prove that there exist integers
qJ
i+1, 1 j 7/x-i, such
that & = lcm(^_
i + 1
,.. . , ^ + 1 ) and
r„-«l«Ui and ^ ± l
e P
( [ J L ] )
for 1 j 7M_i and 1 i fi. Here [x] denotes the largest integer less than or
equal to x. In particular, this implies that lcm(S') = q^ G P(ri).
For i = 1, we have #i = lcm(S^). Since
|#MI = 5M 7M_i/M,
we can divide S^ into 7^-1 disjoint sets of size at most l^. So
By definition, for every s G 5M7-, 1 j 7^-1,
r , |
S
a n d A
e P
(
[
^
]
)
So an application of rule B yields
-i-lcm(S
w
. ) = *
M
lcm(ls,, ) ^ f l ) C P ( [ -
! L
- ] ) , 1 J 7„-i-
' / L I 1 ' £ 4
X
' H '
X
' / L i 1
7
(Here for a set i? = {61,62,... , 6S}, /c£? = {/c6i, ^62,... , fc6s}.)
If we define
q^ = lcm(S,Mi) for 1 j 7 ^ 1 ,
we have proved the claim for i = 1.
Assume next that the claim holds for i = k. So there exist integers q^_fc+15
1 j 7/u-fc, such that
^ - l c m ( ^ _ ,
+ 1
, . . . , ^
+ 1
) ,
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