THE RELATION OF THE PERIOD MODULO
TO THE RANK OF APPARITION OF m IN THE FIBONACCI SEQUENCE
JOHN VINSON, AEROJET-GENERAL CORPORATION, SACRAMENTO, CALIF.
The Fibonacci sequence i s defined by the r e c u r r e n c e relation,
(1)
V
2
=
V
l
+
%
,
n = 0, 1, 2,
and the initial values u 0 = 0 and VLt = 1. Lucas [ 2 , pp. 297-301] h a s shown that
every i n t e g e r , m 5 divides some m e m b e r of the sequence, and also that the sequence
i s periodic modulo m for every m. By this we mean t h e r e i s an integer, k, such
that
(2)
uk+
Definition.
tive integer,
= u (mod m) ,
n = 0, 1, 2, *** .
The period modulo m, denoted by s ( m ) , i s the s m a l l e s t p o s i -
k, for which the s y s t e m (2) i s satisfied.
Definition.
The rank of apparation of m , denoted by f(m), i s the s m a l l e s t
positive integer, k, for which u, = 0 (mod m).
Wall [3] has shown that
(3)
u n = 0 ( m o d m ) iff
f(m)|n.
In 1p a r t i c u l a r , since u , . = uun = 0 (mod m); we have
'
s(m)
(4)
f(m) |s(m) .
Definition.
We define a function t(m) by the equation f(m)t(m) = s(m).
We note that t(m) i s an integer for all m. The purpose of this paper i s to
give c r i t e r i a for the evaluation of t(m).
Now we give some r e s u l t s which will be needed l a t e r .
(5)
v
'
2
n
u
• = u u
+ (-1)
n-1
n n-2 v ;
This can be proved by induction, using the r e c u r r e n c e relation (1).
This paper was p a r t of a t h e s i s submitted in 1961 to Oregon State University in p a r t i a l
fulfillment of the r e q u i r e m e n t s for the degree of m a s t e r of A r t s .
37
THE RELATION OF THE PERIOD MODULO m TO THE RANK
38
/^
(6)
u
<?n - (3n
,
1 + N/5
, .
= ——-f1—
- , w
here # = —+
5 V5 and, £>„
w n o T O ru — •
n
a - j8
[April
1 - \/5
This is the well-known "Binet f o r m u l a . " It gives a natural extension of the Fibonacci
sequence to negative values of n.
By using the relation a (3 = (-1) , we find
u
(7)
_„n
=
1
(* -)
n+1
u.S i
F r o m this we see that the r e c u r r e n c e (1) holds for the extended sequence.
By solving the s y s t e m
ak - /3 k = (a - (3) u k
a • ak - /3 . /3 k = (or - 0) u k + 1
k
for a
k
and ft , we obtain
or
= u.k+1
/3u k = (1 - £ ) u k + u k _ x = , u
k +
uk_x
and
/3 k =, u k + 1 - « u k = (1 - a) u k + u k _ x = £ u k + u k - 1
Then
{a
- 0 , ^ = «nk+r - ^
= (.u,+
V l
, V - (^uk +
By expanding and recombining we get (for n > 0)
%k+r = X I .UkVlVj
Now if we set k = f(m). we find
(8)
v ;
u r, x, = Up. . , u (mod m)
nf(m)+r
f(m)-.l r
V l )V
1963 ]
OF APPARITION OF m IN THE FIBONACCI SEQUENCE
39
We note that this is valid for negative as well as non-negative i n t e g e r s ,
= u
r.
L e m m a 1. t(m) is the exponent to which u f , . - belongs (mod m).
n
Proof: Suppose u f / . ., = 1 x(mod m).
Then from v(8)
we have u -, x,
;
;
-^
f(m)-l
nf(m)+r
(mod m) for all r . It follows from the definition of s(m) that s(m) _< nf(m)
and thus u f ,
,_-. = 1 (mod m) implies t(m) = s(m)/f(m) <_ ne
Now we s e t r = 1 and n = t(m) in (8) to obtain
t(m)
uf
'
f(m)-l
Thus
uf,
-, ,
-, x
= 1 1 , / .-, v, -, = u , ,, .. = u t = 1 (mod m) .
x
v
;
t(m)f(m)+l
s(m)+l
t(m) i s the s m a l l e s t positive n for which u nf , , = 1 (mod m),
that i s ,
._ 1 belongs to t(m) (mod m).
T h e o r e m 1.
F o r m > 2 we have
i) t(m) = 1 or 2 if f(m) i s even, and
ii) t(m) = 4 if f(m) is odd.
A l s o , t(l) = t(2) = 1.
Conversely,
t(m) = 4 implies f(m) is odd, t(m) = 2 i m -
plies f(m) i s even, and t(m) = 1 implies f(m) is even or m = 1 or 2.
Proof.
The c a s e s
m = 1 and m = 2 a r e easily verified.
Now suppose
m > 2 and s e t n = f(m) in (5) to get
U
f(m)-1 -
u
f(m)uf(m)-2
+
("1)f(m) -
( " 1 ) f ( m ) (modm)
.
If f(m)
i s even we have u?, x _. = 1 (mod m ) , and i) follows from L e m m a 1.
v
'
t(m)-l
'
If f(m)
i s odd we have uS, x -. = - 1 (mod m )/ ?, and since m > 2 , u | , . 1
v
'
f(m)-l
' l(m)-l
^ 1 (mod m). This implies u f , ,_-. ^ ± 1 (mod m) and then
3
2
i
Up, x *- = u_p. . _. u„. , _, = -Up, , ., r ± ± 1 v (mod m).
f(m)-l
f(m)-l f(m)-l
f(m)-l
'
Finally,
u4
= (u2
)2 = (-1) 2 = 1 (mod m) and, by L e m m a 1,
t(m) = 4.
The c o n v e r s e follows from the fact that the c a s e s in the direct statement of
the t h e o r e m a r e all inclusive.
T h e o r e m 2. Let p be an odd p r i m e and let e be any positive integer. Then
i)
ii)
t(p e ) = 4 if 2 | f(p),
t(p e ) = 1 if 2 | f(p) but 4 1 f(p),
iii)
t(p e ) = 2 if 4 | f(p), and
iv)
t(2 e ) = 2 for e 2 3 and t(2) = t(2 2 ) = 1 .
40
THE RELATION OF THE PERIOD MODULO m TO THE RANK
[April
Conversely, if q r e p r e s e n t s any p r i m e , then t(q ) = 4 implies f(q) is odd,
e
a
e
t(q ) = 2 i m p l i e s 4|f(q) or q = 2 and e _>. 3, and t(q ) = 1 implies 2jf(q) but
4 | f(q) or q e = 2 or 4.
Proof. Wall [3, p. 527] has shown that if p n + 1 \ u
then f(p n + 1 )= pf(p n ).
•It follows by induction that f(p ) = p f(p), where k is some non-negative integer.
e
We emphasize that f(p ) and f(p) a r e divisible by the s a m e power of 2, since this
fact i s used s e v e r a l t i m e s in the sequel without further explicit r e f e r e n c e .
e
e
In c a s e i), f(p ) i s odd and the r e s u l t is given by setting m = p in T h e o r e m 1.
e
e
In Gc a s e s ii) a n d i i i ) , f(p ) i s even and we may set m = p , n = 1, and
r - | f ( p ) in (8) to get
UI-P/ ep\
|f(p )
2
-
U
4T/ P\e 1
f(p )-l
u
H o, «
-|f(pe) x
v (modP
pe)
'
which, in view of (7), i s the s a m e as
U
u
f(pe)-1
s
if(pe)
2
Now |f(p e ). = p-p^(p),
where
l f ( P e/ ), i+u1 „
. (-D2iVF
/™Hr^
<mod
if(pe)
^
k i s s o m e non-negative integer, and we see that
f(p) I
if(pe). Then from (3) we have p | u l f , e so that we may divide the above
2
2XvP )
congruence by u l f , y We get
i,
uA . ^ ^
f(p6)Now in c a s e ii),
.= ( - l ) 2 f ( P e ) + 1
(modpe)
~f(p) i s odd and so is H{pe)f
2
<*
and the l a s t congruence gives
e
u f / ^x - = • 1 (mod pe) and t h u s , by L e m m a 1, t(p ) = 1 .
ilp^J-l
In c a s e iii) the congruence b e c o m e s
u
f(pe)-i
=
-1
( m o d P e ) '.
since
Hip)
a r e both even.
and
if(pe>
Then
V)-!
and by L e m m a 1 again,
t(p e ) = 2.
s
l (modpe)
.
1963 J
OF APPARITION OF m IN THE FIBONACCI SEQUENCE
In c a s e iv) we can easily verify t(2) = t(2 2 ) = 1.
41
That t(2 e ) = 2 for e ^ 3
follows from r e s u l t s given by Carmichael [1, p. 42] and Wall [ 3 , p. 527].
These
results are, respectively:
A.
Let q be any p r i m e and let r be any positive integer such that (q, r) = 1.
, X+l I
,-,
X+a
If q u and q
/ u , then q
u
_ and q X+a+1 / u
_ except when q = 2
n
H
L
I n
A
I nrqa
/] n r q a
and X = 1.
B. Let q be any p r i m e and let X be the l a r g e s t integer such that s(q )
e-X
= s(q). Then s(q e ) = q
s(q) for e > \ .
xl
The hypotheses of A. a r e satisfied by q = 2, X = 3, and n = f(2 3 ), and
->3+a| u
.
a|
T4-.cn
e
/OV4.U 4. .c/o3+ax
we find that 2
I v.f/o3\ ^ff 20 T|k- It follows from (3) that f(2
) m u s t be a m u l t i '\f(23)
pie of f(2 3 ),
for e ^ 3.
hence f ( 2 3 + a ) = 2 a f(2 3 ).
Since f(23) = 2f(2) we have f(2 e ) = 2 e " 2 f(2)
Now set q = 2 and X = 1 in B.
We get s(2 e ) = 2 e _ 1 s ( 2 ) . Thus for
e ^ 3 we have
t(2e} =
3(2!) = f^m
f(2 e )
2e
= 2
.
f(2)
The converse follows from the fact that the c a s e s in the d i r e c t statement
the t h e o r e m a r e all inclusive.
of
This completes the proof,
Now we give a l e m m a which is needed in the proof of the next theorem.
L e m m a 2.
If m has the p r i m e factorization
m = qt
i) s(m) = 1. c. m.
l^i^
ii)
f(m) = 1. c. m.
l<i^n
. . . qr
n
,
then
( s ( q . 1 ) } , and
(ffq1)}
1
Wall has given i).
relatively p r i m e ,
q2
m|u,
The proof of ii) is as follows: Since the q. 1 a r e p a i r w i s e
i s equivalent to q. i u, (i = 1, 2, 0°* , n), which, by (3),
i s equivalent to f (q. i ) k (i = 1, 2, •«• , n).
The s m a l l e s t .positive k which s a t i s -
fies these conditions i s
k = 1. c. m.
l<i<n
which, according to the definition of f(m),
(ffa?1)}
1
,
gives the d e s i r e d r e s u l t .
42
THE RELATION OF THE PERIOD MODULO m TO THE RANK
[April
T h e o r e m 3. We have
i)
ii)
t(m) = 4 if m > 2 and f(m) i s odd.
t(m) = 1 if 8/f m and 2 | f (p) but 4 f f (p) for every odd p r i m e ,
divides m ,
iii)
p,
which
and
t(m) = 2 for all other
m.
Proof: F r o m what has already been given in T h e o r e m 1, we see that it suffices to show that the conditions given h e r e in ii) a r e both n e c e s s a r y and sufficient
for t(m) = 1.
Let m have the p r i m e factorization
f(q^) = 2 V
w h e r e the K. a r e odd i n t e g e r s .
6
at
t(q
Then
s^1)
x
(i = 1, 2, -
m = q11q22
, n)
q n
m
and set
,
By T h e o r e m 1, we may set
i
) = 2
(i = 1, 2, ••• , n)
= f (q?i )t (q?i) = 2
1
1
where 6. = 0, 1, or 2.
K. (i = 1, 2, ••• , n).
F r o m L e m m a 2 we
have, where K i s an odd integer,
, x
1. c. m. r / # i \ i
0 max(7i + &\ )xr
w
s(m) = - , ' . „
isfa-Mr = 2
i
i 'K,
L
J
'
l<i<n
l
.
1. c. m.
p,^.
,
£/
0max^iT/
f(m) = , .
ffa- 1 )
= 2
iK, and
x ;
VH ;
5
l£i<n
i
4., ^
i X/-P/ x
0max(7-+<5 ) - m a x 7 x
t(m) = s(m)/f(m) = 2
i i'
i
Now suppose t(m) = 1.
Then max (7. + 5.) = m a x / . .
7, <L 7i
+ 5,
Let y, = m a x / . . We have
< m a x (7 + 6.) = max 7, = 7u ,
6
and thus <5k = 0 and t ( q Q'
£ Kkv) = 2~ k = 1. It follows from T h e o r e m 2 that 4 ( f ( q k k ) ,
that i s , that y^ < 1.
Then for all i,
6. < max (7. + 6.) = max 7. = 7 ,
Furthermore,
<1 .
6. = 1 is i m p o s s i b l e , for 5. = 1 is the s a m e as t(q. i ) = 2 which
i m p l i e s , by T h e o r e m 2, that
2 f(q. *)
and thus 7. > 1.
Then we would have
1963]
/. 1 + t
OF APPARITION OF m IN THE FIBONACCI SEQUENCE
1
> 2j3 which
is c o n t r a r y to max (:>'
. +•£.) < 1.
z.
" I I 7
^1
id t CX'
and t ( q . 1 ) = 2 = l s
in ii).
43
Thus for all
i,
61
'
which, by T h e o r e m 2, is equivalent to the conditions given
Now suppose, conversely, that the conditions given in ii) a r e satisfied, which,
as we have just seen, is equivalent to the condition t ( q . x ) = 1 for all i.
s(q?i) = f(qj'i) t ( q ^ )
Then
for all i.
Then L e m m a 2 gives
. ,
S(m
1. c. m.
r
. a\ . -,
1. c . m .
' = l<i<_n { S <V>} = l<i<n
rp/^ixi
^ t(q i
}
'' =
P.
.
f(m)
and thus t(m) = s(m)/f(m) = 1.
Our l a s t t h e o r e m is of r a t h e r different c h a r a c t e r . Once again, we need a p r e liminary l e m m a .
L e m m a 3.
Let p be an odd p r i m e .
i)
f(P) | (p - 1)
if
Then
p = ±. 1 (mod 10) ,
ii)
f(p) | (p + 1)
if
p = ± 3 (mod 10) ,
iii)
s(p) | (p - 1)
if
p = i l (mod 10) , and
iv)
s ( p ) | ' ( p + 1) but s(p) | 2(p+ 1) if p SE+ 3 (mod 10).
Lucas [2, p. 297] gave the following r e s u l t :
P
| U p - l if
P
" ~
X
^mod
10
)
and
P I UD+1
if
P = - 3 (mod 10).
We get i) and ii) by applying (3) to this result, Wall [3. p. 528] has given iii) and iv).
T h e o r e m 4.
i)
ii)
Let p be an odd p r i m e and let e be any positive integer. Then
t(p e ) = 1 if p = 11 or 19 (mod 20),
t(p e ) = 2 if p = 3 or 7 (mod 20),
iii)
t(p e ) = 4 if p = 13 or 17 (mod 20), and
iv)
t(p e ) 4 2 if p =• 21 or 29 (mod 40).
Proof: T h e o r e m 2 shows that t(p ) is independent of the value of e,
i s sufficient to consider e = 1 throughout the proof.
hence
THE RELATION OF THE PERIOD MODULO m TO THE RANK
44
[April
so that by
Fermatrs
belongs to t(p) (mod p), it follows that t(p) j(p-l).
Now if
If follows from the definition of ff(p) that p j( u f , .
theorem,
u
Then, since u f . .
1
f(p)-l
~
X (mod
p)
'
p = 3 (mod 4) we have 4 \ (p - 1) and thus t(p) ^ 4.
i)
H e r e p = 3 (mod 4)
4 | f(p).
so t(p) ^ 4.
Suppose t(p) = 2.
Now p = ± 1 (mod 10) and, by L e m m a 3 i ) ,
4 | (p - 1).
Then, by T h e o r e m 2,
f(p)
(p - 1)
But this is impossible when p = 3 (mod 4), hence
and
thus
t(p) ^ 2 and we
m u s t have t(p) = 1.
ii) Again p = 3 (mod 4) and t(p) ^ 4.
Also p = ±. 3 (mod 10) and it follows from
L e m m a 3 that s(p) ^ f(p) and t(p) = s(p)/f(p) ^ 1.
Hence t(p) = 2.
iii) We have just seen that t(p) ^ 1 when p = ± 3 (mod 10), which i s h e r e the c a s e .
Also,
f(p) J (p + 1).
Now p = 1 (mod 4) so that
and it follows from T h e o r e m 2 that t(p) ^ 2.
iv) Suppose t(p) = 2.
= t(p)f(p) = 2f(p)).
t(p) = 2 i m p l i e s
Then by T h e o r e m 2,
Furthermore,
8( (p - 1).
4 j (p + 1)
and thus
f(p),
Hence t(p) = 4.
4 | f(p) and thus
s(p) | (p - 1)
since
8 js(p)
(since
s(p)
p = ± 1 (mod 10). Then
But we have p - 1 = 20 or 28 (mod 40)
p - 1 = 4 (mod 8), so that 81 (p - 1) i s impossible.
which
gives
Hence t(p) ^ 2.
We naturally ask if a ^ t h i n g m o r e can be said about t(p ) for
29 (mod 40).
4 |
p = 1, 9, 2 1 ,
The following examples show that the t h e o r e m is "complete' 1 :
p =
1 (mod 40)
p =
9 (mod 40)
p = 21 (mod 40)
p = 29 (mod 40)
t(521) = 1,
t(809) = 1,
t(101) = 1,
t(29) = 1,
t(41) = 2,
t(761) = 4
t(409) = 2,
t(89) = 4.
t(61) = 4.
t(109) = 4.
Now we might ask whether t h e r e is a number, m, for which t(p ) is
d e t e r m i n e d by the modulo m r e s i d u e c l a s s to which
this question i s not known.
p belongs.
always
The answer to
We note that the p r i n c i p l e s upon which the proof of
T h e o r e m 4 is b a s e d a r e not applicable to other moduli.
1963]
OF APPARITION OF m IN THE FIBONACCI SEQUENCE
45
REFERENCES
1.
n
n
R. D. C a r m i c h a e l , On the n u m e r i c a l factors of the arithmetic forms a ± (3 ,
Ann. of Math. 15 (1913-1914), 30-70.
2.
E. Lucas, Theorie des fonctions numeriques simplement periodiques,
3.
J. Math. 1 (1878), 184-240, 289-321.
D. D. Wall, Fibonacci s e r i e s modulo m, A m e r . Math. Monthly 67 (I960), 525-532.
^wywyYwvwvvvv^^
Amer.
SPECIAL NOTICE
The Fibonacci Association has on hand 14 copies of Dov J a r d e n ,
Recurrent
Sequences, Riveon L e m a t e m a t i k a , J e r u s a l e m , I s r a e l . This i s a collection of p a p e r s
on Fibonacci and Lucas n u m b e r s with extensive tables of factors extending to the
385th Fibonacci and Lucas n u m b e r s .
The volume s e l l s for $5.00 and is an e x c e l -
lent investment. Check or money o r d e r should be sent to V e r n e r Hoggatt at San J o s e
State College, San J o s e , Calif.
REFERENCES TO THE QUARTERLY
Martin G a r d n e r , Editor, Mathematics G a m e s , Scientific A m e r i c a n ,
J u n e , 1963
(Column devoted this i s s u e to the helix. )
A Review of The Fibonacci Quarterly will appear in the Feb.
1963 i s s u e of the
Recreational Mathematics Magazine.
FIBONACCI NEWS
B r o t h e r U. Alfred r e p o r t s that he is c u r r e n t l y offering a one unit c o u r s e
on
Fibonacci N u m b e r s at St. M a r y ' s College,
Murray B e r g ,
Oakland, Calif. , r e p o r t s that he has computed phi to
2300 decimals by dividing F l l o 0 4 by F 1 1 0 0 3 on a computer.
some
Any inquiries should
be a d d r e s s e d to the editor.
Charles R. Wall, Ft. Worth, T e x a s , r e p o r t s that he is working on his m a s t e r ' s
t h e s i s in the a r e a of Fibonacci r e l a t e d topics.
SORTING ON THE B-5000 - - Technical Bulletin 5000-21004P S e p t . , 1961,
Burroughs Corporation, Detroit 32, Michigan.
This contains in Section 3 the use of Fibonacci n u m b e r s in the merging of information using t h r e e tape units instead of the usual four thus effecting considerable
efficiency.
(This was brought to our attention by Luanne Angle my e r and the pamphlet
was sent to us by Ed Olson of the San J o s e office.)
HAVE YOU SEEN?
46
[April
S. L. Basin, Generalized Fibonacci Numbers and Squared Rectangles, A m e r i c a n
Mathematical Monthly, pp. 372-379, April, 1963.
J . A. H. Hunter and J. S. Madachy s Mathematical D i v e r s i o n s , D. Van Nostrand
Company, I n c . , P r i n c e t o n , New J e r s e y , 1963.
•S. K. Stein, The Intersection of Fibonacci Sequences,
Michigan Math.
Journal,
9 (1962), Dec. , No, 4 a pp. 399-402. (Correction)
L. Carlitz, Generating Functions for P o w e r s of Certain Sequences of N u m b e r s ,
Duke Math. J o u r n a l , Vol. 29(1962), D e c , No. 4, pp. 521-538. (Correction)
V. E. Hoggatt, J r . and S. L. Basin, The F i r s t 571 Fibonacci N u m b e r s ,
Recrea-
tional Mathematics Magazine, Oct. , 1962, pp. 1 9 - 3 1 .
A. F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions,
The
A m e r i c a n Mathematical Monthly, M a r . , 1963, pp. 289-291.
S. L. Basin, The Appearance of Fibonacci Numbers and the Q - m a t r i x in E l e c t r i cal Network Theory, Math. Magazine, Vol. 36, No. 2, March 1963, pp. 84-97.
J. Browkin and A. Schinzel, On the Equation 2
- D = y 2 , Bulletin de l'Academie
Polonaise des Sciences, Serie des sci. math., a s t r , et phys. —Vol. VIII. No. 5,
I960, pp. 311-318.
Georges Browkin and Andre Schnizel, Sur les n o m b r e s
triangulares,
de Mersenne qui
Comptes rendues des s e a n c e s de l'Academie
des
sont
sciences,
t. 242, pp. 1780-1781, seance du 4 Avril 1956.
C. D. Olds,
Continued F r a c t i o n s ,
Random House (New Mathematical
Library
S e r i e s - - p a r t of the Monograph P r o j e c t of SMSG) 1963.
This is an excellent understandable t r e a t m e n t of the subject at a reasonable
level with many i n t e r e s t i n g topics for those devoted to the study of i n t e g e r s
with special p r o p e r t i e s .
L. Zippin, Uses of Infinity7, Random House (New Mathematical L i b r a r y S e r i e s , 1962.)
This has no index which, makes the Fibonacci topics h a r d e r to find but t h e r e
a r e s e v e r a l i n t e r e s t i n g comments t h e r e .
Mannis Charosh, P r o b l e m Department, Mathematics Student J o u r n a l , May, 1963.
In the editorial comment following the solution of P r o b l e m 187, t h e r e is a
little generalized r e s u l t s i m i l a r to problem B-2 of the E l e m e n t a r y P r o b l e m s
and Solutions section of the Fibonacci Q u a r t e r l y , Feb. , 1963.
A. Rotkiewicz, On Lucas Numbers with Two Intrinsic P r i m e D i v i s o r s ,
Bulletin
de l'Academie Polonaise des Sciences, Serie des sci. math. , a s t r . et phys. —
Vol. X. , No. 5, 1962.