RESIDUES OF GENERALIZED BINOMIAL
COEFFICIENTS MODULO A PRODUCT OF PRIMES
ERIK R. TOU
Honors Mathematics Thesis
Department of Mathematics and Computer Science
Gustavus Adolphus College
St. Peter, MN
Spring 2002
ABSTRACT.
Take a sequence
u 
n
of positive integers generated by u 0  0 , u1  1 ,
u n  au n 1  bu n 2 ( n  2 ). For a  b  1 , u n  becomes the familiar Fibonacci sequence. From
this sequence, one may define the generalized factorial [ n]!  u1u 2 u n and the generalized
binomial coefficient C m, n   [m  n]! [m]![n]! . The residues of these coefficients modulo a
prime have a number of well-documented properties, including a remarkable self-similarity. While
the result is more complicated when these coefficients are reduced by product  of primes, some
of the modulo p properties are inherited in the modulo  case.
1. INTRODUCTION
Define the binomial coefficient Bm, n as
 m  n  m  n !
 
Bm, n   
.
m! n!
 n 
A theorem of E. Lucas [5] gives a way to determine the residue of a binomial coefficient modulo
a prime p in terms of the base-p digits of m and n. If m   m j p j and n   n j p j , where
0  m j , n j  p , then

Bm, n   B m j , n j
 mod p.
This theorem also allows for considerable flexibility in the evaluation of the congruence. For
example, given any positive integer  ,

Bm, n    B m j , n j

j 0



   B m j , n j   B m j , n j 
 j 
 j 






 B m  p , n  p  B m mod p  , n mod p  ,





 

where m  p  is the integer quotient of m by p  , and m mod p  is the remainder. A generalized
binomial coefficient corresponds to a particular sequence u n  , with which n! is replaced by the
product u1u2 un . When u n  is a second-order recursive sequence, a more limited version of
Lucas’ theorem will hold.
In both cases, the array of residues has a self-similar structure modulo p. The situation modulo
a product  of primes is more complex, and possesses fewer of the properties of the binomial
and generalized binomial coefficients modulo p. However, some of these properties do translate
from primes to products of primes, which  inherits from its prime factors.
2. SOME RESULTS REGARDING THE GENERAL FIBONACCI SEQUENCE u n 
Definition 1. Let a, b  Z , and let p be any prime. The sequence u n  is defined recursively as
u0  0 , u1  1 , and un  au n1  bu n2 for n  2 .
In the case a  b  1 , u n  is the Fibonacci sequence. When a  2 and b  1, u n  is the
sequence of non-negative integers.
Definition 2. Let r denote the rank of apparition of p in the sequence u n  ; that is,
r  r  p   min n  N : un  0 mod p  . Let t  t  p  denote the least period of the sequence
un mod p of residues, if it exists. Let s  s p  t r .
Throughout this paper, consider the prime p and the integers a and b to be fixed, and assume
that a and b are not both zero.
Recall that there exists a closed form for the generalized Fibonacci sequence. Let  and 
be the roots of the quadratic equation x 2  ax  b  0 . That is, let  
a  a 2  4b
2
and  
a  a 2  4b
2
.
Then, for any n  0, 1, 2,  , u n     . For convenience, we denote D  a 2  4b . In the case
n
n
that D  0 , the closed form is un  n  n1 . With this description, we can extend a number of
well-known identities of Fibonacci numbers to the more general case.
Lemma 1. For n  1 ,
1
un   
 2
n 1

12 n1 n 

j 0



 a n2 j 1 D j ,
2
j

1


where 12 n 1 represents the greatest integer less than or equal to 12 n 1 .
Proof. The proof of this identity consists largely of an algebraic manipulation of the closed form
already given. Expanding  and  , we have
n n 1
 
un 
 
 2
n

 
n
 a D  a D
a D

 a 2 D
2


Using the binomial theorem on the a  D
expanded, as follows:

n


n
 

.


and a  D


terms, this equation may be further

n
n n
i
i
 
 1   1  n  n  ni
D    a ni  D 
   a
  
 2   D  i 0  i 
i 0  i 


n
i
 n
 n
 1   1 
i
  
   a ni D 1   1    a ni
 2   D  0in  i 
0i  n  i 
i even
 i odd
 
n



D 1   1  .


 
i
i

In the second of the two summations, i is even, and so 1   1i  1  1  0 . Thus, the entire sum
reduces to zero as well. In the first summation, 1   1i  1  1  2 . With this fact, and a simple
re-indexing, the expression becomes
1 n1 n 
n
 1   1  2
a n2 j 1
 2   
  

 2   D 
j 0  2 j  1
1
 
 2
n1

12 n1 n 

j 0

2 j 1 

 D



j

a n2 j 1 a 2  4b ,
2
j

1


where 12 n 1 represents the greatest integer less than or equal to 12 n 1 . □
Lemma 2. For n  1 , un1un1  un   b 
2
Proof.
.
First, suppose that D  a2  4b  0 . Then  b 
un  na2 
n 1
, we have
un 1un 1  n  1a2 
n2

n 1

 n2  1 a2 
 n2 a2 
2n 1
2n 1
 a2 
 un 2   b n 1 .
2n 1
n  1a2 n
a2
4
, and using the closed form
Next, suppose D  0 . Then, the closed form is u n     , and we use the following identity:
n
 b
n 1


 a 2  a 2  4b
 
4




 a D a D 

 

4


n
n 1


n 1
  n 1 .
With this, the identity will follow from direct algebraic manipulation:
2
 n   n 
   n 1
un   b   

   
 2n  2 n   2n    2  n 1


   2
   2
n 1
2



 2n  2 n     2  n 1   2n
   2


 2n   n 1 2   2  2   2   2n
   2
 2 n   n 1 n 1   n 1 n 1   2 n

   2
 n 1   n 1  n 1   n 1

 un 1un 1 . □
   2




At this point, a theorem of Fermat’s is stated for future reference, and will be used frequently
from here on.
Theorem 1. (Fermat) Let p be any prime, and let a be any integer. If p does not divide a, then
a p1  1 mod p . Equivalently, we have a p  a mod p .
Lemma 3. Let p be any prime. If p does not divide b, then p divides at least one of u p 1 , u p ,
and u p 1 .
Proof. First, suppose that p  2 . Then, u p  u2  a and u p 1  u3  a 2  b . If a is even, then
2 | u2 . If a is odd, then since 2 does not divide b (by hypothesis), a2  b  1  1  0 mod 2 .
Thus, 2 | u3 . And so, the lemma holds for the case p  2 .
Next, let p be an odd prime and suppose that p | a 2  4b . Then,
Fermat’s theorem may be applied in conjunction with Lemma 1 to get
u p  2 p 1u p
12  p 1  12  p 1 and

1
2
 p 1

j 0


 p  p  2 j 1 2

 a
a  4b
 2 j  1
Recall that for k  1, 2, , p  1 ,
 

mod p .
j
   0 mod p; as a result, this sum reduces to
p
k

1  p 1
 p 0 2
  a a  4b 2
 p

 a 2  4b

1
2
p 1
mod p .
By assumption, p | a 2  4b . Thus, p | u p .
Lastly, suppose that p is an odd prime but that p does not divide a 2  4b . Using Lemma 2 and
the previous case in this lemma,
u p 1u p 1  u p   b 
2
p 1

 a 2  4b

p 1
  b  p 1
mod p .
Since p divides neither a 2  4b nor  b , Fermat’s theorem tells us that
a
2
 4b

p 1
  b  p 1  1  1  0
mod p .
Hence, p divides at least one of u p 1 and u p 1 . In general, then, if p does not divide b, then p
must divide at least one of u p 1 , u p , and u p 1 . □
Corollary 1. If p does not divide b, then r  p   p  1.
Corollary 2. Suppose that the following statements hold:
(1) p1, p2 , , pk are primes, with p1  p2    pk ,
(2) it is not the case that p1  2 and p2  3 ,
(3) p1 does not divide b.
Then, if there exist non-negative integers 1, 2 , , k such that r  p1  p11  r  p2  p22  
 r  pk  pk k , we must have i  0 for all i  2, 3, , k .

Proof. For i 2, 3, , k, since gcd p1 1 , pi

i
1
and r  p1  p1 1  r  pi  pi i , it follows that


pi i | r  p1  . That is, there exists a positive integer v such that vpi i  r  p1  . By Corollary 1
above, r  p1   p1  1  pi . This results in the following:
pi i  vpi i  r  p1   p1  1  pi .
Therefore, i  0 . □
Also, note that if p1  2 and p2  3 , the same argument may be applied to give i  0 for
i 3, 4, , k.
3. GENERALIZED BINOMIAL COEFFICIENTS MODULO p
Definition 3. Given a generalized Fibonacci sequence and a non-negative integer n, define the
bracket factorial as [n]! u1  u 2   u n , with [0]! 1 . Then, for any pair m, n of non-negative
integers, the generalized binomial coefficient is defined as
C m, n  
[m  n]!
.
[n]![m]!
If any factors are zero, then the zeroes in the numerator and denominator are to be cancelled in
pairs. A result by Holte [3] guarantees that the number of zero factors in the numerator will
either be one greater than the number of factors in the denominator or will be exactly equal.


 
Definition 4. A prime p is said to be ideal for the sequence u n  if r p k 1  p  r p k for all
positive integers k.
Lemma 4. Let p be an ideal prime for the sequence u n  , and let r  r  p . Suppose that for
some  0, 1, 2, , m  m'rp   i and n  n'rp   j , where m' and n' are non-negative
integers, and 0  i, j  rp  . Then, i  j  rp  implies


C m, n   C m'rp   i, n'rp   j  0
mod p .
Proof. As proven in [4] and treated in [2], the number of times that p divides C m, n is the
number of carries that occur when m and n are added in the mixed-radix system with radices
b1 : r  p , b2 : r p 2 r  p  , b2 : r p 3 r p 2 , etc. So, consider the addition i  j in such a
manner. We have the following:
   
 
k
r pk
bk
Carries
Augend
Addend
 
i
+ j
i j
Sum
 1 …
 1 

 1
 2
0
j
j 1
…
…
…
…
…
d  1
d
d  1
…
rp
p
c
0
rp
p
-i
rp
p
-i 1
3
rp 2
p
-i3
j3
2
rp
p
-i2
0
1
*
-i0
j2
1
r
r
-i1
j1
d3
d2
d1
d0
j0
Since 0  i, j  rp  and i  j  rp  , we know that by the above adding scheme, d  1  0 , and
hence c  0 . And so, if we add m  m'rp   i and n  n'rp   j by the same method, there will
be at least one carry from the  th column to the   1st column, because of i and j. There may
be other carries before or after this one, but such an occurrence is not necessary. This one carry
guarantees that p will divide C m, n at least once. Therefore, C m, n  0 mod p . □
In simpler terms, Lemma 4 says that if a particular Cm, n value lies in the lower right
triangle of a matrix of coefficients, with dimension rp  (   0, 1, 2, ... ), then it is a multiple of p.
Theorem 2. ( rp  -step recurrence modulo p) If p does not divide b, and p is an ideal prime for
the sequence u n , we have that for any  0, 1, 2,  and for all m, n  Z  ,





C rp   m, rp   n  ur 1mr  C rp   m, rp   n  1  ur 1nr  C rp   m  1, rp   n

mod p .
Proof. (by induction on  ) First, let   0 . Then for all m, n  Z  , the congruence reduces to
Cmr, nr   ur 1 mr  Cmr, n  1r   ur 1 nr  Cm  1r, nr  ,
which is proven in [3]. Next, consider an arbitrary   0 , and suppose that our congruence holds
for all j  0, 1, 2, ,   1 and all m, n  Z  . Then,

 

C rp   m, rp   n  C rp  1   pm , rp  1   pn  .
Applying the inductive hypothesis, we have that


C rp  1   pm , rp  1   pn 
 ur 1 
 ur 1 

 C rp 
pmr
pnr
 C rp
 1
1

 pn mod p  .
 pm, rp  1   pn 1
  pm 1, rp
 1
( )
By Corollaries 1 and 3 in [3], we know that for any positive integer k, u r 1k  u kr 1  0 mod p .
Thus, Fermat’s theorem tells us that ur 1 p  ur 1 mod p  . As a result, (  ) reduces to


C rp  1   pm , rp  1   pn 

 C rp 

 ur 1 mr  C rp  1  pm, rp  1   pn  1
 ur 1 nr
1
  pm 1, rp  1  pn

(  )
mod p .
The method here is to apply the inductive hypothesis to the above equation, ultimately repeating
the process p times. Applying the inductive hypothesis to (   ), we have
ur 1 mr  Crp  1  pm, rp  1   pn 1 ur 1 nr  Crp  1   pm 1, rp  1  pn
 ur 1 mr  ur 1 mr  C rp  1  pm, rp  1   pn  2


 ur 1 mr  ur 1  pn1 r  Crp  1   pm  1, rp  1   pn  1


 ur 1 nr  ur 1  pm1 r  Crp  1   pm  1, rp  1   pn  1
 ur 1 nr  ur 1 nr  Crp  1   pm  2, rp  1  pn
(1)
(2)
(3)
(4)
The general binomial coefficient in expressions (2) and (3) can rewritten as


C rp   m  1   p  1rp  1 , rp   n  1   p  1rp  1 .
 p  1rp  1   p  1rp  1  2 p  1rp  1  p  rp  1  rp  , Lemma 4 dictates
C rp  1   pm  1, rp  1   pn  1  0 mod p , and hence the entire expression reduces to
Since
u    Crp
 u    C rp
r 1
mr 2
r 1
 1
nr 2

 pm, rp  1   pn  2
 1
that
(5)

  pm  2, rp  1  pn .
(6)
When this process is repeated with (5) and (6), the result will have expressions of the forms (1),
(2), (3), and (4). Each time, the expressions corresponding to (2) and (3) will be congruent to
zero modulo p. To see this, consider an arbitrary ith iteration of this process ( 0  i  p ); the
general binomial coefficient corresponding to that in expression (2) will be as follows:


C rp  1   pm  i , rp  1   pn  1

 C rp  m  1   p  i rp

 1

, rp   n  1   p  1rp  1 .
Again,  p  i rp  1   p  1rp  1  2 p  i  1rp  1  p  rp  1  rp  (as long as i  p ), whence
Lemma 4 tells us that this coefficient is congruent to zero modulo p. A similar method will
demonstrate the same result for the general binomial coefficient that corresponds to that in
expression (3) of the ith iteration. Hence, if we apply the inductive hypothesis p times to (  ), we
will accumulate factors of the form ur 1 mr and u r 1 nr in front of the general binomial
coefficients in the corresponding expressions (1) and (4), respectively, giving us the following:

C rp   m, rp   n

   Crp
  Crp
 u
r 1

 pn
mr p
 1
 pm, rp  1   pn  p 
nr p
 1
  pm  p , rp  1
 ur 1
(7)
Simplification yields the following result:

 
C rp   m, rp   n  ur 1
  Crp
mr p

 
 m, rp   n  1  ur 1

Again using Fermat’s theorem, we have that ur 1 p
may be further simplified to get



  Crp
nr p
  u 
mr

1 mr
r 1


 m  1, rp   n

mod p .
mod p. Hence, expression (7)
C rp   m, rp   n  ur 1mr  C rp   m, rp   n  1  ur 1nr  C rp   m  1, rp   n

mod p . □
Example 1: Fibonomial Coefficients Modulo 2 and 3. Using the Fibonacci sequence, with
p  2 or 3, and m, n  Z  , C12m, 12n  C12m, 12n 1  C12m  1, 12n mod p . To see
this, first take p  2 . Then, applying Theorem 2 with r  r 2  3 and   2 , we have the
following:

 
 
C 12m, 12n   C 3  2 2  m, 3  2 2  n
 F4 3m  C12m, 12n  1  F4 3n  C12  m  1, 12n .
Since F4 3  33  1 mod 2 , this reduces to
C12m, 12n 1  C12  m 1, 12n .
For p  3 , note that F5 4  54  1 mod 3 ; applying Theorem 2 with r  r 3  4 and   1
will similarly yield C12m, 12n  C12m, 12n 1  C12m  1, 12n mod 3 .
Example 2: Fibonomial Coefficients Modulo 6. Using the Fibonacci sequence, and given
m, n  Z  , C12m, 12n  C12m, 12n 1  C12m 1, 12n mod 6 . For simplicity, we use the
following notation: c  C12m, 12n ; c1  C12m, 12n 1 ; c2  C12m 1, 12n . Under this
notation system, we want to show that c  c1  c2 mod 6 . As seen in Example 1, c  c1  c2 
mod 2 and c  c1  c2  mod 3 . That is, the integer c  c1  c2  is a multiple of both 2 and 3.
This means that c  c1  c2  is also a multiple of 6, which ultimately tells us that c  c1  c2
mod 6. Or, using the original notation, C12m, 12n  C12m, 12n 1  C12m 1, 12n
mod 6.
4. GENERALIZED BINOMIAL COEFFICIENTS MODULO A PRODUCT OF PRIMES
Given the considerable body of knowledge concerning the generalized binomial coefficients
modulo a prime, the natural tendency when considering these coefficients modulo a product of
primes is to extend previous knowledge (to what extent it is possible) to this more general case.
If one is given the residues of a particular Cm, n value modulo primes p1 , p2 , , pk , the most
direct way to find the residue value modulo p1  p2  pk is to make use of the Chinese
Remainder Theorem, which is restated here.
The Chinese Remainder Theorem.1 Let m1 , m2 ,, mk denote k positive integers that are
relatively prime in pairs, and let a1 , a2 ,, ak denote any k integers. Then the congruences
n  ai mod mi , i  1, 2, , k , have common solutions. Any two solutions are congruent modulo
m  m1  m2  mk .
Fortunately, the Chinese Remainder Theorem is constructive. By defining xi to be the solution
to the congruence m mi xi  1 mod pi  , i  1, 2, , k , the common solution is
k
m
x ja j .
m
j
j 1
w0  
1
Given in [6], page 31. See the text for more details on this theorem and the Corollary to follow.
Corollary. Given primes p1 , p2 , , pk such that p1  p2    pk , let   p1  p2  pk . Let
xi be the solution to the congruence relation  pi xi  1 mod pi  , for each i  1, 2, , k .
Lastly, taking n   , let ni  n mod pi . Then,
k

i 1
pi
n  w0  
xi ni mod   .
Proof. This follows directly from the proof of the theorem, and in particular the definition of
w0 . Here, we take mi  pi and ai  ni for each i  1, 2, , k . Then, the theorem gives
w0  i 1  pi xi ni as the common solution, where the xi are defined as the solutions to the
k
congruences stated above. Thus, for any j  1, 2, , k ,
k

i 1
pi
w0  
xi ni 

pj
x j n j  n j  n mod p j  .
That is, n  w0 mod pi  for each i  1, 2, , k . Therefore, n  w0 mod   . □
Definition 5. Given primes p1 , p2 , , pk , ( p1  p2    pk ), we can use the corollary above
to construct an operator f : Z k  Z , for which f z1 , z 2 ,, z k   i 1  pi xi zi . Note that
k
when zi  n mod pi ( i  1, 2, , k ), f reduces to the congruence relation given by the corollary;
that is, n  f n1 , n2 ,, nk  mod   .
Lemma 5. Suppose that for integers z1 , z 2 ,, z k , n  zi mod pi  for each i  1, 2, , k . Then,
f z1 , z 2 ,, z k   f n1 , n2 ,, nk   n mod   .
Proof. Let z1 , z 2 ,, z k be as described above. Then,
f z1 , z 2 ,, z k   f n1 , n2 ,, nk 

 j 1 p
k

 j 1 p
k
j
x j z j   j 1
k


pj
x jn j

xj zj  nj .
j
Since n j  n  z j mod p j  , the integer z j  n j is a multiple of p j , for each j  1, 2, , k . And
since  p j is a multiple of all the primes except p j , each term in the summation above is a
multiple of  . Hence, f  y, z   f n1 , n2   0 mod   . Recall that by definition, f n1 , n2 ,, nk 
 n mod   . Therefore, f z1 , z 2 ,, z k   f n1 , n2 ,, nk   n mod   . □
Lemma 6. Suppose that for integers z1 , z 2 ,, z k , there exists an integer a such that zi  azi ' for
each i  1, 2, , k . Further suppose that n  zi  azi ' mod pi  . Then, f z1 , z 2 ,, z k 
 a  f z1 ' , z 2 ' ,, z k '  n mod   .
Proof. Using Lemma 5, we have the following chain of congruences:
k

j 1
pj
n  f z1 , z 2 ,, z k   
k

j 1
pj



k

j 1
pj
xj a z j '  a
 a  f z1 ' , z 2 ' ,, z k '
xjzj
xjzj'
mod   . □
Definition 6. We may also extend the function f to a matrix operation. Given matrices
M 1 , M 2 ,, M k of dimensions d  d, define f M 1 , M 2 ,, M k  componentwise; that is,
f M 1 , M 2 ,, M k   [ f M 1 i, j , M 2 i, j ,, M k i, j ] , 0  i, j  d .
Notation. Let m be a non-negative integer. For any prime pi and any positive integer  , define
the following:
ri  r  pi 
si  t i r i
ti  ti  p   the least period of u n mod pi 
mi '  m  ri
m0,i  m mod ri
m'  m 
m0  m mod 
mi " mi ' mod si
Theorem 3. (Holte) If pi does not divide b, then for m, n  0 ,
Cm, n  Bmi ' , ni 'H mi ", ni " m0,i , n0,i  mod pi  ,
where H mi ", ni " is an r  r matrix determined by the values mi " and ni " , as described in [3].
Furthermore, we will use the convention H mi " , ni "   Hmi "  modsi , ni "   modsi , where 
and  are any pair of non-negative integers. Since mi " mi ' mod si , this means that H mi " , ni " 
 H mi ' , ni '  .
Definition 7. Given a prime pi , let H \ pi
mi ', ni '
denote the associated ri  ri matrix determined
by the values mi ' and ni ' , as described in Theorem 3. Then, given any non-negative integer  ,
define
H \ pi
 
mi ', ni '
 [ B ,   H \ pi
mi '  , ni ' 
] , 0   ,  pi  .
Note that
H \ pi
 
mi ', ni '
has dimensions ri pi   ri pi  , and that
H \ pi
0 
mi ', ni '
 H \ pi
mi ', ni '
.
Additionally, given any pair  ,  of non-negative integers, 0   ,   ri pi  , we denote the
 ,  entry of the matrix H \ pi
H \ pi
 
mi ', ni '
 
mi ', ni '
as
 ,    B  ri ,   ri 
H \ pi
mi '   ri , ni '    ri 
 mod ri ,  mod ri  .
With this we may restate Theorem 3 under a modified system of notation: if pi does not
divide b, then, for m, n  0 , C m, n  Bmi ' , ni ' H \ pi
mi ', ni '
m0,i , n0,i  mod pi  .
Definition 8. For any non-negative integer m, any prime pi , and any non-negative integer i ,


define  m  m  m mod ri pi i and  mi '   m  ri . Note that if i  0 ,  mi '  mi ' .
Theorem 4. Let p1 , p2 , , pk be primes, with p1  p2    pk , and let   p1  p2  pk .
Suppose there exist non-negative integers 1 , 2 , , k such that r1  p11  r2  p2 2  
 rk  pk k (denote this value  ). If for each i  1, 2, , k , pi does not divide b, then for
m, n  0 ,
C m, n   Bm' , n'S m ', n ' m0 , n0 
where

 
mod   ,
 
S m', n'  f H \ p1  1m ',  n  ' , H \ p2  2m  ',  n  ' , H \ p3
1
1
2
2
m', n'
, , H \ pk
m', n'
mod  .
 
Also, if it is not the case that p1  2 and p2  3 , then H \ p2  2m  ',  n  '  H \ p2
2
2
m', n '
as well.
Proof. Let pi be an arbitrary prime, and suppose that i  0 . Then, Theorem 3 tells us that
C m, n  Bmi ' , ni ' H \ pi
mi ', ni '
m0,i , n0,i  mod pi  .
(1)
Now, recall that we have assumed there exist non-negative integers 1 , 2 , , k such that
r1  p11  r2  p2 2    rk  pk k   . Applying Lucas’s theorem to the binomial term, we have
that
Bmi ' , ni '  Bm  ri , n  ri 


 B m  ri pi i , n  ri pi i B m  ri mod pi i , n  ri mod pi i

 Bm   , n   B m  ri  mod pi i , n  ri  mod pi i
Along with expression (1) above, this gives us


mod pi  .


C m, n   Bm   , n   B m  ri  mod pi i , n  ri  mod pi i H \ pi
mi ', ni '
m0,i , n0,i  mod pi  .
Expanding the notation, the last two factors of this expression become


B m  ri  mod pi i , n  ri  mod pi i H \ pi
 Bm mod    ri , n mod    ri 
 H \ pi
mri , nri
mi ', ni '
m mod ri , n mod ri 
(2)
m mod  mod ri , n mod  mod ri  mod pi  .
Next, we show that if i  0 , m  ri  m  m mod    ri  m mod   ri  . This involves writing m
in a mixed-radix system, as in Lemma 4. Taking m as d d i 1d i d 2 d1d 0
m  ri  d d i 1d i d 2 d1
 d d i 1d i 0 i 1 01

 d d i 1d i d 2 d1d 0
R
, we have that
R
 d i 1 d 2 d1
R
R
R
 d i 1 d 2 d1d 0
 m  m mod    ri  m mod   ri  .
R
 r  m mod   r 
i
i
Making this substitution in (2), we get
Bm mod    ri , n mod    ri 
 H \ pi mm mod r m mod r , nn mod r n mod r  m mod   mod ri , n mod   mod ri  .
i
i
i
i
By Definition 7, this is
 
H \ pi mi m mod r , n  n mod r m mod  , n mod   .
i
i
 
Hence, if i  0 , Cm, n  Bm' , n' H \ pi  im  ', n  ' m0 , n0  mod pi  .
i
i

In the case i  0 , we have the simplified relation ri  ri pi i   . Thus, mi '  m' , m0,i  m0 ,
and Theorem 3 gives
C m, n  Bmi ' , ni ' H \ pi
 Bm' , n' H \ pi
m ', n '
mi ', ni '
m0,i , n0,i 
m0 , n0 
 
 Bm' , n' H \ pi  im  ',  n  m0 , n0 
i
i
mod pi  .
Furthermore, Corollary 2 to Lemma 3 tells us that for i  3, 4, , k , i  0 . For these values of i,
we will write C m, n   Bm' , n' H \ pi
m ', n '
m0 , n0  .
Next, we use the matrix version of the
operator f in conjunction with Lemma 6 to conclude that
C m, n  Bm' , n'

 
 
 f H \ p1  1m ',  n  ' , H \ p2  2m ',  n  ' , H \ p3
1
1
2
2
m', n'
, , H \ pk
m', n'
 mod   ,
or more simply,
C m, n   Bm' , n'S m ', n ' m0 , n0 
mod   .
Additionally, if it is not the case that p1  2 and p2  3 , then Corollary 2 tells us that
 
H \ p2  2m  ',  n  '  H \ p2
2
2
m', n '
as well. □
Example: Fibonomial Coefficients Modulo 6. Using the Fibonacci sequence, take p1  2 and
p2  3 . Then, we have the relation   r 2  2 2  3  2 2  12  4  3  r 3  31 ; thus, 1  2 and
2  1 . A simple congruence is C 3,8  B0, 0S 0, 0 3, 8  1 1  1 mod 6 . A more complicated
case is the coefficient C53,42 . Here,  53  53  5  48 , and  42  42  6  36 . Then,
C 53,42   B4, 3S 4, 3 5, 6

 

 

 5  f  H \ 2 5,6, H \ 3 5,6
 5  f H \ 2 16, 12 5,6, H \ 3 12, 9 5,6
2
1
2
0, 0
1
0, 1
 5  S 0, 1 5,6  5 1  5 mod 6 .
5. OPEN QUESTIONS
Modulo p, one may reduce the subscripts of the H \ p
m ', n '
matrix modulo s. This reduction
allows for a complete tabulation of the number of distinct matrices of this type. In the example
above, we were able to derive the relation S 4, 3  S 0, 1 . Since s6  2 , the relation is really
S 4, 3  S 4 mod2, 3 mod2  S 0, 1 . In terms of the Fibonacci sequence, Halton [1] has proven that for any
   
 
positive integer q  p11 p2 2  pk k , r q   lcm r p11 , r p2 2 , , r pk k . A similar result
holds for t q  . A determination of sq  in these terms would allow for a reduction of the
subscripts of S m ', n ' modulo s, and give a way to compute the precise number of distinct S m ', n '
matrices. Modulo p, (where p does not divide b), Holte has found this number to be  p  1s 2 .
Modulo   p1 p2  pk , we can at least say that the number of distinct non-zero S m ', n ' matrices
is no more than
 j1  p j  1s j 2  1 1 .
k
Additionally, any similar results regarding powers of primes would allow the work done here
to expand into the further realm of all positive integers, rather than simply a product of primes.
Recall that the Chinese Remainder Theorem applies to a collection m1 , m2 ,, mk of positive
integers that are relatively prime in pairs. If such a result does exist for powers of primes, it may
be possible to extend Theorem 4 to include these cases as well.
In more general terms, a classification of all prime products with the r1  p11  r2  p2 2  
 rk  pk k relation would delineate the full applicability of the results for the product  .
REFERENCES
[1] Halton, John H., On the Divisibility Properties of Fibonacci Numbers, Fibonacci Quart. 15
(Oct. 1966), 217-240.
[2] Holte, John M., Asymptotic Prime-power Divisibility of Binomial, Generalized Binomial,
and Multinomial coefficients, Transactions of the American Mathematical Society. Vol. 349, No.
10 (Oct. 1997), 3837-3873.
[3] Holte, John M., Residues of Generalized Binomial Coefficients Modulo a Prime, Fibonacci
Quart. 38 (June-July 2000), 227-238.
[4] Knuth, D. E. and H. S. Wilf, The Power of a Prime That Divides a Generalized Binomial
Coefficient, J. reine angew. Math. 396 (1989), 212-219.
[5] Lucas, E., Théorie des functions numériques simplement périodiques, Amer. J. Math. 1
(1878), 184-240.
[6] Niven, Ivan, and Herbert S. Zuckerman. An Introduction to the Theory of Numbers, 3rd ed.
New York: John Wiley & Sons, Inc., 1964.