DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS

Transcription

1 DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto INTRODUCTION I [2], Webb ad Parberry study the divisibility properties of the Fiboacci polyomial sequece {f (x)} defied by the recursio f +2 (x) = xf + 1 (x) + f (x); f 0 (x) = 0, f t (x) = 1. As oe would expect, these polyomials possess may properties of the Fiboacci sequece which, of course, is just the itegral sequece {f (1)}. However, a most surprisig result is that f (x) is irreducible over the rig of itegers if ad oly if p is a prime. I cotrast, for the Fiboacci sequece, the coditio that be a prime is ecessary but ot sufficiet for the primality of f (1) = F. For istace, F 19 = 4181 = I the preset paper, we obtai a series of results icludig that of Webb ad Parberry for the more geeral but clearly related sequece {u (x,y)} defied by the recursio u + 2 (x,y) = xu + 1 (x,y) + y u ( x, y ) ; u 0 (x,y) = 0, u t (x,y) = 1. The first few terms of the sequece are as show i the followig table: u (x,y) 0 1 x 3 4 x 3 x 2 + y + 2xy x 4 + 3x 2 y + y 2 x 5 + 4x 3 y + Sxy 2 x 6 + 5x 4 y + 6x 2 y 2 + y 3 8 x 7 + 6x 5 y + 10x 3 y 2 + 4XV 3 113

2 114 DIVISIBILITY PROPERTIES [April The basic fact that we will eed is that Z [ x, y ], the rig of polyomials over the itegers, is a uique factorizatio domai. Thus, the greatest commo divisor of two elemets i Z [ x, y ] is (essetially uiquely) defied. Useful Property A: if a,j3, ad y are i Z[x,y] ad y afi with y irreducible, the y\a or y /3. For simplicity, we will frequetly use u i place of u (x,y) ad will let / x x + N/ x 2 + 4y a = a(x,y) = g * ad P o = j3(x,y) ot \ = - \/x ~ 2 + 4y * x 2. BASIC PROPERTIES OF THE SEQUENCE Agai, as oe would expect., may properties of the Fiboacci sequece hold for the preset sequece. I particular, the followig two results are etirely expected ad are easily proved by iductio. Theorem 1. For ^ 0, Theorem 2. For m ^ 0 ad ^ 0, u = g a - js u,,_. = u 1 1 u J 1 + y Ju u m++1 m+1 +1 m The ext result that oe would expect is that (u, u + - ) = 1 for ^ 0. To obtai this we first prove the followig lemma. Lemma 3. For > 0, (y, u ) = 1. Proof. The assertio is clearly true for = 1 sice \x t = 1. Assume that it is true for ay fixed iteger k ^ 1. The, sice V i = x u k + y \ - i the assertio is also true for = k + 1, ad hece for all ^ 1 as claimed. We ca ow prove

3 1974] OF GENERALIZED FIBONACCI POLYNOMIALS 115 Theorem 4. For ^ 0, (u N, u,-) = ' Proof. Agai the result is trivially true for = 0 ad = 1 sice u 0 = 0, % = 1, ad u 2 = x. Assume that it is true for = k - 1 where k is ay fixed iteger, k ;> 2, ad let d(x,y) = (u,, u, - ). Sice V i = x \ + y \ - i this implies that d(x,y) u. -y. But (d(x,y), y) = 1 by Lemma 3 ad so d(x,y) u,. But the d(x,y) 1 sice (u,, u, ) = 1 ad the desired result holds for all ^ 0 as claimed. Lemma 5. For ^ 0, u (x, y) r(-l)/2] -lj/2] X ^ / - i - 1 \ -2i-l i Proof. We defie the empty sum to be zero, so the result holds for = 0. For = 1, the sum reduces to the sigle term 0) xu y u = 1 = u t Assume that the claim is true for = k - 1 ad = k, where k ^ 1 is fixed. The Vi = x \ + y Vi [(k-l)/2] v [0^2)/2] = v ( k -1 - * )x k - 2 y + 2 ( k " i' 2 ) x k " 2 i - v + 1 i=0 ^ ' i=0 V ' [ ( k - l ) ^ ],. V2l E ( k -;- l ) x t - 2 v + ; ( t - 1 i - 1 i ), k - v [k/2] ^ * / k - i \ k-2i i i=0 x ' Thus, the result holds for = k + 1 ad hece also for all ^ 0 as claimed.

4 116 DIVISIBILITY PROPERTIES [April 3. THE PRINCIPAL THEOREMS Theorem 6. For m ^ 2, u u if ad oly if m. J ; ' m ' ' Proof. Clearly J u u. Now suppose that u u. m ' m ^ m ' km where k ^ 1 is fixed. The, usig Theorem 2, (k+l)m km+m = u. u M + Jyu. - u km m+1 km-1 m But, sice u u, by the iductio assumptio, this clearly implies that u u,, ^. Thus, u u if m. m ' ' Now suppose that m ^ 2 ad that u u. If m/j, the there exist itegers q ad r with 0 < r < m, such that = mq + r. Agai by Theorem 2, we have that u = u mq+r = u,., u + Jyu u. mq+1 r mq r - 1 Sice u u by J the first part of the F proof, this implies F that u u, _. u. But, sice m i mq * m ' mq+1 r (u, u -) = 1 by Theorem 4, this implies that u u ad this is impossible, sice H 4 I u is of lower degree tha u i x. Therefore, r = 0 ad m ad the proof is complete. Theorem 7. For m ^ 0, ^ 0, (u, u ) = \i / x. m (m,) Proof. Let d = d(x,y) = (u, u ). The it is immediate from Theorem 6 that U (m,) ' Now, it is well kow that there exist itegers r ad s with, say, r > 0 ad s < 0, such that (m,) = r m + s. Thus, by Theorem 2, rm (m,)+(-s) 1/ v U, - -r y u / \ -, U J (m,) -s+1 (m,)-l -s But the d u ad d u by Theorem 6 ad so d u, xu,-. But, (d, u,-, )= 1 -s ' rm J ' (m,) -s+1 -s+1 1 by Theorem 4, ad so d u, \ by Useful Property A from Sectio 1. Thus, d = u, x as claimed. (m,)

5 1974] OF GENERALIZED FIBONACCI POLYNOMIALS 117 Theorem 8. The polyomial u = u (x,y) is irreducible over the ratioal field Q if ad oly if is a prime. Proofo From Lemma 5, if we replace y by y 2 we have u (x,y*) = ] > ] ( " J " X ) * - 2U1 y 2i which is clearly homogeeous of degree - 1. Now it is well kow (see, for example, [l, p. 376, problem 5]) that a homogeeous polyomial f (x, y) over a field F is irreducible if ad oly if the correspodig polyomial f (x, 1) is irreducible over F. Sice u (x, 1) is irreducible by Theorem 1 of [ 2], it follows that u (x,y2) ad hece also u (x,y) is irreducible over the ratioal field ad thus is irreducible over the itegers. 4. SOME ADDITIONAL THEOREMS For the Fiboacci sequece { F }, for ay ozero iteger r there always exists a positive iteger m such that r F. Also, if m is the least positive iteger such that r I F, the r F if ad oly if m. It is atural to seek the aalogous results for the sequece of Fiboacci polyomials {f (x)} cosidered by Webb ad P a r b e r r y ad the geeralized sequece {u (x,y)} cosidered here. I a sese, the first problem is solved by Webb ad Parberry for the sequece of Fiboacci polyomials, sice they give explicitly the roots of each such polyomial. However, it is still ot clear exactly which polyomials r(x) possess the derived property. O the other had, it is immediate that the first result metioed above does ot hold for all polyomials r(x). For example, if c is positive, o liear factor x - c ca divide ay f (x) sice this would imply that f (c) = 0, ad this is impossible sice f (x) has oly positive coefficiets. Alog these lies, we offer the followig theorems which, amog other thigs, show that the secod property metioed above does hold without chage for u (x,y) ad hece also for f (x). We give this result first. Theorem 9. Let r = r(x, y) be ay polyomial i x ad y. If there exists a least positive iteger m such that r u, the r u if ad oly if m. Proof. By Theorem 6, if mj, the u u. Therefore, if r u we have by trasitivity that r u. Now suppose that r u ad yet m. The there exist itegers q ad s with 0 < s < m such that = mq + s. Therefore, by Theorem 2, u = u mq+s = u,., u mq+1 s + J yu u _.. mq s-1 Sice r 1 u mq ad r u, it follows that r! i ' u,., u. But (u, u,_,) =! ad this mq+1 s mq mq+1 implies that r u. But this violates the miimality coditio o m ad so the proof is complete.

6 118 DIVISIBILITY PROPERTIES [April Theorem 10. For ^ 2, ' 1 / kir \ u (x,y) = i x - 2 W y c o s J Proof. From the proof of Theorem 8, it follows that Vx.y 2 ) = y -\f f. A = y " lf m where f (x) is the Fiboacci polyomial metioed above. Thus, u (x,y) = y ( _ 1 ) / 2 f ( x / ^ 7 ) ad it follows from [2, page 462] that f (x/ -Jy) = " (-5_ - 2i cos i^l ) k=i v ^ J This, with the precedig equatio, immediately yields the desired result. Corollary 10. For ^ 2, eve, (-2)/2 / k \ u (x,y) = x II I x 2 + 4y cos 2 I k=i \ / ad, for odd, ( " l ) / 2 / ktt \ u (x,y) = ( x 2 + 4ycos 2 J k=l \ / Proof. This is a immediate cosequece of Theorem 10, sice, for 1 ^ k < / 2, k ( - k)77 cos = - cos - It is clear from the precedig theorems that there is a precise correspodece betwee the polyomial factors of u (x,y) ad those of u (x, 1) = f (x). Thus, it suffices to cosider

7 1974] OF GENEEALIZED FIBONACCI POLYNOMIALS 119 oly those of f (x). Also, it is clear that, except for the factor x, the oly polyomial factors of f (x) with itegral coefficiets cotai oly eve powers of x. While we are ot able to say i every case which eve polyomials are factors of some f (x) we offer the followig partial results. Theorem 11. (i) x J f (x) if ad oly if is eve. (ii) (x 2 + 1) f (x) if ad oly if 3. (iii) (x 2 + 2) f (x) if ad oly if 4. (iv) (x 2 (v) (x 2 + 3) f (x) if ad oly if 6. + c) f (x) if c / 1, 2, or 3 ad c is a iteger. Proof. Sice, except for x oly, all polyomials with itegral coefficiets dividig ay f (x) must be eve, the results (i) through (iv) all follow from Theorem 9 with y = 1. Oe has oly to observe that f 2 (x) is the first Fiboacci polyomial divisible by x, that f 3 (x) is the first Fiboacci polyomial divisible by x 2 + 1, ad so o. Part (v) follows from the fact that 1 < 4 cos 2 a < 4 for a a i the iterval (0, TT/2). Theorem 12. Let m be a positive iteger ad let N(m) deote the umber of eve polyomials of degree 2m ad with itegral coefficiets which divide at least oe (ad hece ifiitely may) members of the sequece {f (x)}. The Proof. y = 1 that N(m) < m I r 1 4 k Let f(x) be ay polyomial couted by N(m). It follows from Corollary 10 with / x 2m, 2m-2,, 2. f(x) = x + a m _ i x " ' a i x m = (x 2 + a.) J 3=1 where a. = 4 cos 2 j3. with 0 < 3. < TT/2 for each j. Therefore, 0 < a < 4 for each j. Sice a k is the k elemetary symmetric fuctio of the tf. r s, it follows that a o 0 < a < m-k ( m ty ad hece that

8 120 DIVISIBILITY PROPERTIES, p OF GENERALIZED FIBONACCI POLYNOMIALS as claimed. N(m) < (") f r 1 4 k Of course, the estimate i Theorem 12 is exceedigly crude ad ca certaily be improved. It is probably too much to expect that we will ever kow the exact value of N(m) for every m. Our fial theorem shows that with but oe added coditio the geeralizatio to u (a,b) of the first result metioed i this sectio is valid. Theorem 13. Let r be a positive iteger with (r,b) = 1. The there exists m such that r u (a,b). 1 m Proof. Cosider the sequece u (a,b) modulo r. Sice there exist precisely r 2 distict ordered pairs (c,d) modulo r, it is clear that the set of ordered pairs {(u 0 (a,b), u^a.b)), (u^a.b), u 2 (a,b)),, (u 2<a,b), u r (a,b))} must cotai at least two idetical pairs modulo r. That is, there exist s ad t with 0 < s < t ^ r 2 such that ad But ad ad this implies that u (a,b) = u,(a,b) (mod r) s z u (a,b) = u t + 1 (a,b) (mod r) bu s _ 3 (a,b) = u g + 1 (a,b) - au g (a,b) bu, 1(a,b) = u, - (a,b) - au,(a,b) bu (a,b) = bu,..(a,b) (mod r) Sice (r,b) = 1, this yields u 1(a,b) = u t _ x (a,b) (mod r). Applyig this argumet repeatedly, we fially obtai so that r u f ' C S 0 = u (a,b) = u, (a,b) (mod r) s-s t-s (a,b) ad the proof is complete. REFERENCES 1. S. MacLae ad G. Birkhoff, Algebra, The MacMilla Compay, New York, N.Y., W. A. Webb ad E. A. Parberry, "Divisibility Properties of Fiboacci Polyomials," Fiboacci Quarterly, Vol. 7, No. 5 (Dec. 1969), pp