98 ir r e d u c ib il it y o f l u c a s a n d g e n e r a l iz e d l u c a s p o l y n o m ia l s [f eb . if p is a p rim e , th en p is a facto r of...

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1.

INTRODUCTION

In [5] , Webb and P a r b e r r y discuss several divisibility properties for the sequence \ F (x)} of Fibonacci polynomials defined recursively by (1)

F0(x) = 0,

Fife) = 1,

F n + 2 (x) = xF n + 1 (x) + F n (x),

n ^0 .

In particular, Webb and P a r b e r r y prove that F (x) is irreducible over the integral domain of the integers if and only if p is a prime. In [l] , Bergum and Kranzler develop many relationships which exist between the s e quence \ F (x) } of Fibonacci polynomials and the sequence \L (x)} of Lucas polynomials d e fined recursively by (2)

L0(x) = 2,

L

Ljfe) = x 5

n+2

(x)

=

xL

n+l(x)

+ L

n(x)'

n

~° '

Specifically, Bergum and Kranzler show that (3)

L (x) I L

(x)

iff

m = (2k - l)n,

k ^ 1.

With n = 1, we see that X | L (X) for all odd integers m so that the result of Webb and P a r b e r r y does not hold for the sequence (L (x)}. In [4] , Hoggatt and Long show that the result of Webb and P a r b e r r y does hold for the sequence {u (x,y)} of generalized Fibonacci polynomials defined by the recursion

(4)

U 0 (x 5 y) = 0,

Ui(x,y) = 1,

u

n+

2(x'y)

=

xU

n+l(x'y)

+ yU

n

(x y)

'

'

n

- ° -

The purpose of this paper is to obtain n e c e s s a r y and sufficient conditions for the i r reducibility of the elements of the sequence {L (x)} as well as the elements of the sequence {v fe,y)} of generalized Lucas polynomials defined by the recursion (5)

V 0 fe,y) = 2,

Vi(x,y) = x,

V n + 2 fe,y)

= xV n + 1 fe,y) + yV n (x 5 y),

The first few t e r m s of the sequence { V fe,y)} are 95

n > 0.

96

IRREDUCIBILITY OF LUCAS AND GENERALIZED LUCAS POLYNOMIALS n

[Feb.

V (x,y) n J

1

X 2

2

x + 2y

3

x 3 + 3xy

4

x 4 + 4x 2 y + 2y2

5

x 5 + 5x3y + 5xy2

6

x 6 + 6x4y + 9 x V + 2V3 x 7 + 7x5y + 14x3y2 + 7XV3

7 8

8

x + 8x6y + 20x4y2 + 16x2^

+ 2V4

9

x 9 + 9x7y + 27x5y2 + SOxV + 9XV4

Observe that L (x) = V (x,l) so that with Jy = 1, we also have the first nine t e r m s n n of the sequence { L (X)} . 2. IRREDUCIBILITY OF L (x) nN The basic fact that we shall use is found in [2, p. 77] and is Theorem 2.1. (Eisenstein's irreducibility criterion.) F(x) = a x n

+a

.. x ~ n-1

For a given prime p,

let

+ • • • + a

be any polynomial with integral coefficients such that a

n—x

= a 9 = • • • = a 0 = 0 (mod p ) , \\—u

a n

i

0 (mod p ) ,

a 0 £ 0 (mod p 2 )

then F(x) is irreducible over the field of rationals. To establish our first irreducibility theorem, we use the following. Lemma 2.1. Every coefficient of L

2n

(x), except for the leading coefficient, is divisible

by 2 and 4 does not divide the constant term. Proof. If n = 1 then L2(x) = x2 + 2 and the lemma is obviously true.

Assume the

lemma is true for n. In [1] , we find L 2 k (x) = L2k(x) - 2 ( - l ) k .

(6) Hence, (7)

L 2 n + 1 (x) = Lj n (x) - 2 . By the induction hypothesis, it is obvious that L

of L

.. (x) is divisible by 2.

2n+l

,-,(x) is monic and every coefficient

F u r t h e r m o r e , since L

2^

(x) has constant t e r m

+2 we see

that L 2 (x) has constant t e r m +4, thus L ^(x) has constant t e r m +2. Therefore, v 2n+1 2n constant term of L .. (x) is divisible by 2 but not by 4 and the lemma is proved. An immediate result of Lemma 2.1 with the aid of Theorem 2.1 is

the

1974]

IRREDUCIBILITY OF LUCAS AND GENERALIZED LUCAS POLYNOMIALS

97

Theorem 2.2. The Lucas polynomial L . (x) is irreducible over the rationalsfor k ^ 1. 2K

Although L (x) is not irreducible if p is a p r i m e , we can show that

L (x)/x is i r -

reducible for every odd prime p. F i r s t we note, as is pointed out in [l] , that L n (x) = an + /3 n ,

(8)

where a = (x + ^x 2 + 4)/2 and |3 = (x - N/X2 + 4 ) / 2 . L n (x) = (x + N;x2 + 4 ) n / 2 n + (x n

Hence, if n = 2m + 1 we have

N/xr+T)n/2n n

* *

k

n k

k/2

x - (a x 2 + 4 ) = 2 I > . x i r i r v + 4 r - + > 11 K-D^X (X2 + 4)

\k=o

k=o \

'

/

(9) = 2

k=0 \ m

/ k

ti:(i)(^

(n 1

= 2- - >WL?W k U n - 2s 2 2s . k=0 s=0

x

f

x

f

Therefore, m

For each s,

(11)

k

/

\ /

22s,

\

k=0 S=0 X / V / 0 < s < m, we see that the coefficient of x

2

-(n-2s-l)>

, - 11-1

§W0)

n

n = 2m + 1

is

= 2m + l .

k=s

When s = 0, we have the leading coefficient of L (x) which is 1 so that

When s = m in (11), we have the constant term of L (x) which is n. If we now let n be an odd prime p and recall that p divides

U)

98

IRREDUCIBILITY OF LUCAS AND GENERALIZED LUCAS POLYNOMIALS

[Feb.

if p is a p r i m e , then p is a factor of (11) for each value of s,

S

~

2

Hence, by Eisenstein's criterion, the following is true. Theorem 2.3. The polynomials L (x)/x are irreducible over the rationals if p is an odd prime. By (11) and the fact that the coefficients of L (x) are integers, we have Corollary 2.1. If n = 2m + 1 then 2

divides

UW) for any s such that 0 ^ s ^ m. Using (3) together with Theorems 2.2 and 2.3, we have Theorem 2.4.

(a) The ^Lucas polynomials L n (x), n ^ 1, rationals if and only if n = 2 for some integer k — 1.

are irredicuble over the

(b) The polynomials L (x)/x, n odd, are irreducible over the rationals if and only if n is a prime. 3.

IRREDUCIBILITY OF V (x,y)

It is a well known fact that (13)

U (x,y) =

n 0n „ " P , a. -

n s> 0

and V n (x,y) = an + / 3 n ,

(14)

n ^ 0 ,

where a = (x + \/x2 + 4 y ) / 2 and j3 = (x - ^x 2 + 4 y ) / 2 . In [4] , we find Lemma 3.1. (a) For n ^ 0, TT

.

v

U (x y) =

n '

[(n-l)/2] \ ^ / n - k - 11 n-2k-l k

LJ I k=0

X

k

)X

'

(b) For n > 0, m > 0, (U (x,y), U (x,y)) = U, , (x,y) . J J m n ,J (m,n) Using (13) and (14), a straightforward argument yields

y

•

1974]

IRREDUCIBILITY OF LUCAS AND GENERALIZED LUCAS POLYNOMIALS Lemma 3.2.

(a) V ^ x . y ) = y U ^ x . y ) + U n + 1 ( x , y ) , (b)

(C)

U 2 n (x,y) = U n (x,y)V n (x,y), U

2n(x'^V(2k+l)n+l(x>y>

+

n a 1; n 2= 0 ;

^ k - D n ^ V

=

(2k+l)n(x'y)U2a+l(x'y)-

Using (a) of Lemma 3.1 and 3.2, we have, for n — 1,

,, ,

V^

v

=

[(n-2)/2] . . XP n - k - 2 )

2^ \ k=0

k

n-2k-2 k+1 ^

)

*

x

+

y

that

[n/2] , v V""* / n - k \

XJ

'

I k )

k=0

[n/2]

99

X

x

n-2k k

y

'

[n/2]

(15) k=l

x

/

k=0

x

'

[n/2]

E

/n - k - l\

1 k=l \

k-i

n ;

k

x

n-2k k ,

y

+ x

n

•

Hence, Lemma 3.3.

(a) For n ^ 1, V (XjV2) is homogeneous of degree n. (b) If n is odd then x is a factor of V (x,y 2 ) and V (x,y 2 )/x

is h o -

mogeneous of degree n - 1. By (b) of Lemma 3.1, (U

(x,y), U

(x,y)) = 1. Using this fact together with (b) of

Lemma 3.2 and induction on k in (c) of Lemma 3.2, one obtains Lemma 3.4. If k > 1 then V n (x,y) | v ( 2 k - i ) n ( x ' y ) ' In [3, p. 376, Problem 5 ] , we find Lemma 3.5. A homogeneous polynomial f(x,y) over a field F is irreducible over F if and only if the corresponding polynomial f(x, 1) is irreducible over F. Using Lemmas 3.3 and 3.5 with Theorem 2.4, we have 2 Theorem - 3.1. (a) The polynomials Vn (xjV ) are irreducible over the rationals if and only if n = 2 for some integer k ^ 1.

(b) The polynomials V (x,y2)/x, n odd a r e irreducible over the r a tionals if and only if n is an odd prime. Since f(x,y) is irreducible if f(x,y 2 ) is irreducible and x is a factor of V (x,y) for n odd by (15), we apply Lemma 3.4 and Theorem 3.1 to obtain Theorem 3.2. only if n = 2

(a) The polynomials V (x,y) are irreducible over the rationals if and

for some integer k greater than o r equal to one.

100

IEREDUCIBILITY OF LUCAS AND GENERALIZED LUCAS POLYNOMIALS (b) The polynomials V (x,y)/x, n odd,

Feb. 1974

a r e irreducible over the r a -

tio nals if and only if n is an odd prime. Letting y = 1 and n = 2m + 1 in (15), we see that

/..^

T

(16)

L

/ v/

n(x)/x

=

\ " " s n - k - 11 n

2 J k=l

kX

k-1

X

n-2k-l ,

+X

n-1

'

'

Comparing the coefficients of x ~

~

in (16), 1 ^ s ^ m ,

with the result obtained

in (11), we have Corollary 3.1. If n = 2m + 1 and 1 ^ s ^ m then

,-w-»2(iX:) - (n;: il) f • REFERENCES 1. G. B e r g u m a n d A . Kranzler, Linear Recurrences-Identities and Divisibility P r o p e r t i e s , unpublished. 2.

G. Birkhoff and S. MacLane, A Survey of Modern Algebra, The Mac Millan Company, New

3.

G. B i r k o f f a n d S . Mac Lane, Algebra, The Mac Millan Company, New York, N. Y., 1967.

4.

V. E. Hoggatt, J r . , a n d C . T. Long, "Divisibility P r o p e r t i e s of Generalized Fibonacci

York, N Y. , 1965.

Polynomials," Fibonacci Quarterly, to appear April, 1974. 5. W. A. Webb and E. A. P a r b e r r y , "Divisibility P r o p e r t i e s of Fibonacci Polynomials," Fibonacci Quarterly, Vol. 7 (Dec. 1969), pp. 457-463.