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BEGliiEHS 5 CORNER
E d i t e d by D M I T R i
THORO
San J o s e S t a t e C o l l e g e , San J o s e ,
California
THE EUCLIDEAN ALGORITHM I
1. INTRODUCTION
Consider the problem of finding the greatest common divisor of 34 and 144.
The factorizations 34 = 2 • 17, 144 = 24 • 32 make this a trivial problem. However , this approach Is discouraging when one deals with, say., "long" Fibonacci
numbers. Fortunately in Prop, 2 of Book VII, Euclid gave an elegant algorithm.
As usual, we shall designate the g. c, d. of s and t by (s,t).
2. THE ALGORITHM
The algorithm may be defined by the following flow chart.
A ^ B means
A replaces B, i. ee , set B = the current value of A.
M represents the remainder in the division of K by L.
I — K
J —L
W
->' K —L
M—L
/"~"^^^\Yes
M = o? \
> (I,J) = L
->f Stop j
3z:
L— K
Flow Chart for Computing the G. C. D of Positive Integers I and J
53
BEGINNERS 1 CORNER
54
F o r I = 13 and J = 8,
[Feb.
the s u c c e s s i v e values of K, L ,
K
L
M
13
8
5
3
2
8
5
3
2
5
3
2
1
0
1
and M a r e :
The l a s t value of L i s the d e s i r e d g.c.d. In the following computation, (10946,
2584) = the l a s t n o n - z e r o r e m a i n d e r .
4
2584)10946
10336
4
610|2584
2440
4
" 144(610
576
4
34J144
136 4
8|34
324
2|8"
8.
0
In this d i s c u s s i o n we shall e m p h a s i z e computational
considerations.
T h e r e a r e , however, n u m e r o u s " t h e o r e t i c a l ! t applications of the Euclidean A l gorithm.
As LeVeque [1] e x p r e s s e s it,
cative n u m b e r t h e o r y . "
f?
.. . it i s the c o r n e r s t o n e of m u l t i p l i -
F o r a r e l a t e d t h e o r e m s e e Glenn Michael
[ 2 ] , this
issue.
3.
A FORTRAN PROGRAM
With an occasional glance at our flow c h a r t , it is e a s y to decipher the following F o r t r a n p r o g r a m .
( F o r t r a n i s a p r o b l e m - o r i e n t e d language
commonly
u s e d in c o n v e r s i n g with e l e c t r o n i c digital c o m p u t e r s . )
(i)
A = B m e a n s A i s r e p l a c e d by B„
(ii)
The READ and PUNCH s t a t e m e n t s r e f e r to c a r d input/output.
(iii) In this context, N = K / L is an i n s t r u c t i o n to s e t N equal to [ K / L ] ,
i. e. , the g r e a t e s t Integer not exceeding K / L
fixed point quotient).
( s o m e t i m e s called an i n t e g e r o r
Thus if K = 13 and L = 3,
N will equal 4.
(iv) The symbol for multiplication i s an a s t e r i s k .
1964]
BEGINNERS' CORNER
(v)
55
A "conditional transfer' 1 is achieved by using an IF statement: if M
< 0, go to statement 3 for the next instruction; otherwise go to statement 4.
(vi) The FORMAT and END statements are technical requirements (which
may be ignored).
READ 10, I, J
10 FORMAT (315)
K = I
L = J
2 N = K/L
M = K - L *N
IF (M) 3, 3, 4
4K - L
L = M
GO TO 2
3 PUNCH 10, I, J, L
END
4.
Length of the Algorithm
A natural question arises: What is the "length" of this algorithm? I. e. ,
if s and t are given, how many divisions are required to compute (s,t)
via
the Euclidean Algorithm?
Let us designate this number by N(s,t). For convenience we may assume
s > t. Thus for n > 1, N(n + l,n) = 2; the first division yields the remainder
1, whereas the second results in a zero remainder — signifying termination of
the algorithm.
(As a byproduct we see that any two consecutive integers are
relatively prime.)
In Part II we shall see how Fibonacci numbers (Fi = F ? = 1 , F., 1 = F.
1
L
i+i
i
+ F. .,) were used by Lame" to establish a remarkable result. Additional properties of N(s,t) are suggested in the following exercises.
5.
El.
EXERCISES
Note that the Euclidean Algorithm applied to the positive integers s
and t may be described by the equations
BEGINNERS* CORNER
56
r
8 = 1%+ r l 9
0 2= xt < s
t = rtq2 + r2s
0 < r2 < rj
i
+
= ^
r
3»
0 < r3 < r2
= r
, q + r .J
n - 20
n-lTi
n
r . = r q ,n + 0
n-1
n TI+1
r
0 ^ r
n
Explain why we m u s t r e a c h a r e m a i n d e r
b e r of s t e p s .
E2.
[Feb, 1964]
< r
n-1
(r
- ) which i s z e r o In a finite n u m -
Hint: Look at the i n e q u a l i t i e s .
In E l , show that ( s , t ) = r
(the l a s t n o n - z e r o remainder),,
Use r e p e a t e d applications of P r o b l e m 1.3
Hint:
[3],
E3. (a) Verify that M = K - L * N i s the r e m a i n d e r in the division r e p r e s e n t e d by s t a t e m e n t 2 (N = K/L) of the F o r t r a n p r o g r a m .
(b) Can the F o r t r a n p r o g r a m be u s e d to compute (I, J) when I <
E4.
P r o v e that if n ^ 3 , then N(n, 3) = 1, 2, o r 3.
E5.
Suppose that n > 5 i s chosen at r a n d o m .
J?
Find the probability that
N(n,5) > 2.
E6.
P r o v e that for n > 3 ,
E7.
F o r what values of n i s 3 < N(2n - 5,n) ^ 6?
E8 8
Express
E9.
Investigate the following c o n j e c t u r e : If a - F K , then N(n,a) < K - 1.
(F
N(n + 3,n) = 2 5 3, or 4.
.., F ) as a function of n.
Can n be any positive i n t e g e r ?
E10.
number.
Investigate the following c o n j e c t u r e :
Let F ^ 2
b e any Fibonacci
Then m a x N(n, F) = 1 + m a x (n, F - 1).
REFERENCES
1.
William J. LeVeque, Topics in Number T h e o r y , Addison-Wesley, Reading,
M a s s . , 1 9 5 6 , Vol. I, Chap. 2.
2. Glenn Michael, "A New Proof for an Old P r o p e r t y , ! T Fibonacci
Quarterly,
Vol. 2 5 No. 1, F e b . 1964, p . 5 7 .
3.
D. E. Thoro ? "Divisibility I , " Fibonacci Q u a r t e r l y , Vol. 1, No. 1, F e b . 1963,
p. 5 1 .