The University of Sydney
MATH2068
Number Theory and Cryptography
(http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/)
Semester 2, 2008
Lecturer: R. Howlett
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657
Tutorial 3
1.
The Fibonacci sequence is defined by
F
0
= 0,
F
1
= 1 and
F
i
=
F
i

2
+
F
i

1
for
all
i
≥
2. (In Question 3 last week we investigated the sequence (
F
i
) modulo a
prime
p
.)
(
i
)
Suppose that a Fibonacci number
F
n
is divisible by some positive integer
d
,
and write
F
n
+1
=
k
.
Show that
F
n
≡
kF
0
(mod
d
) and
F
n
+1
≡
kF
1
(mod
d
), and use induction on
i
to prove that
F
n
+
i
≡
kF
i
(mod
d
) for all
integers
i
≥
0.
(
ii
)
Use Part (
i
) and induction on
j
to prove that if
d

F
n
then
d

F
jn
for all
natural numbers
j
.
Solution.
(
i
)
Observe that
kF
0
= 0.
But
F
n
≡
0 (mod
d
), since
d

F
n
by hypoth
esis.
Thus
kF
0
≡
F
n
(mod
d
).
And
kF
1
=
k
=
F
n
+1
; so certainly
kF
1
≡
F
n
+1
(mod
d
). This shows that the statement “
kF
i

1
≡
F
n
+(
i

1)
and
kF
i
≡
F
n
+
i
(mod
d
)” is true for
i
= 1. Assume now that
i >
1, and
the statement holds with
i

1 in place of
i
. Thus
kF
i

2
≡
F
n
+(
i

2)
(mod
d
)
kF
i

1
≡
F
n
+(
i

1)
(mod
d
)
and adding these congruences we deduce that
kF
i
=
k
(
F
i

2
+
F
i

1
) =
kF
i

2
+
kF
i

1
≡
F
n
+(
i

2)
+
F
n
+(
i

1)
=
F
n
+
i
.
Since also
kF
i

1
≡
F
n
+(
i

1)
(mod
d
) it follows that the statement holds
for
i
, and hence for all positive integers, by induction.
In particular
kF
i
≡
F
n
+
i
for all natural numbers
i
, as required.
(
ii
)
Since we are given that
d

F
n
, Part (
i
) tells us that
F
n
+
i
≡
kF
i
(mod
d
) for
all
i
, where
k
=
F
n
+1
. In particular, if
F
i
≡
0 (mod
d
) then
F
n
+
i
≡
k
0
≡
0
(mod
d
). That is, for all
i
, if
d

F
i
then
d

F
n
+
i
.
Since
F
0
= 0 we see that
d

F
jn
is true when
j
= 0.
This starts the
induction. Now suppose that
j >
0 and that
d

F
(
j

1)
n
. By what we have
just shown, it follows that
d

F
n
+(
j

1)
n
; that is,
d

Fjn
, as required.
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 One '08
 Howlett
 Number Theory, Cryptography, Natural Numbers, Prime number, FN, kfi, Fibonacci entry point, Fn1 Fn

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