ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints in th singl indpndnt variabls X and Y. Th diophantin problm f(x) = g(y) is strongly rlatd to th absolut irrducibility and th gnus of f(x) g(y ) as pointd out by Davnport Lwis and Schinzl [DLS]: Thorm A. Lt f(x) b of dgr n > 1 and g(y ) of dgr m > 1. Lt D(λ) = disc(f(x) + λ) and E(λ) = disc(g(y) + λ). Suppos thr ar at last [n/2] distinct roots of D(λ) = 0 for which E(λ) 0. Thn f(x) g(y ) is irrducibl ovr th complx fild. Furthr th gnus of th quation f(x) g(y) = 0 is strictly positiv xcpt possibly whn m = 2 or m = n = 3. Apart from ths possibl xcptions th quation has at most a finit numbr of intgral solutions. Th purpos of this not is to handl som spcial cass. For an intgr > 1 w st f (X) = X(X + 1)... (X + 1). For svral scattrd ffctiv and inffctiv rsults on th quation (1) f (x) = f l (y) in intgrs x y w rfr to [BS] [MB] [SS] [SST1] [SST2] and [Sh]. By using an algbraic numbr-thortic argumnt w can guarant th conditions of Thorm A in crtain cass. Lt I dnot th st of intgrs for which f (X) is ithr irrducibl or it has an irrducibl factor of dgr 2. Our conjctur basd upon svral numrical xampls is that I is th whol st of positiv intgrs mor xactly ithr f (X) or (X)/(2X + 1) ar irrducibl dpnding on th parity of. Applying f 1991 Mathmatics Subjct Classification: Primary 11D41. Rsarch supportd in part by th Hungarian Acadmy of Scincs by Grants 16975 19479 and 23992 from th Hungarian National Foundation for Scintific Rsarch. [303]
304 B. Brindza and Á. Pintér Eisnstin s thorm on can s that th prims blong to I and w hav chcd by computr that {1 2 3... 30} I. Thorm 1. If and l ar lmnts of I with 2 < < l thn th polynomial f (X) f l (Y ) is irrducibl (ovr C) and (1) has only finitly many solutions. Morovr som simpl inqualitis lad to Thorm 2. Lt and m b intgrs gratr than 2. Thn th quation ( ) y f (x) = in positiv intgrs x and y m has only finitly many solutions. R m a r. Similar (ffctiv) rsults in th cass = 2 l > 2; = 2 m > 2 and m = 2 > 2 wr obtaind in [Y] and [SST2] rspctivly. Ths quations can b tratd by Bar s mthod. P r o o f o f T h o r m 1. Th discriminant of th polynomial f (X)+λ is dnotd by D (λ) i.. D (λ) = C (f (x) + λ) f (x)=0 (cf. [DLS]) whr C is a non-zro absolut constant. To show that D (λ) and D l (λ) hav no common zros w ta any irrational zros α and β l of f and f l rspctivly and put K = Q(α β l ). Th crucial stp is that instad of th comparison of f (α ) and f l (β l ) w show that thir fild norms with rspct to K ar not qual. If f (X) is irrducibl thn a simpl calculation yilds ( f N K/Q (f (α )) = (0)... f ) [K:Q(α )] (1 ) ; furthrmor if is vn thn f (X) is always divisibl by th linar factor 2X + 1 and in cas I as was pointd out by A. Schinzl w gt ( 2 f N K/Q (f (α )) = (0)... f ) [K:Q(α )] (1 ). ( 1) /2 ( 1)!! According to ths formula for an intgr n > 2 w writ f n(0)... f 1/(n 1) n(1 n) n a n = n if n is odd 2 n f n(0)... f n(1 1/(n 2) n) n n (n 1)!! if n is vn.
Sinc For convninc st b 1 = b 2 = 1 and Irrducibility of som polynomials 305 b = f (0)... f (1 ) ( > 2). f +1(i) = (i + )f (i) i = 0 1... 1 f +1( ) =! w hav th rcursion b +1 = b (!) 2 and thrfor b = (2!... ( 1)!) 2 ( > 2). To prov that th squnc a n n = 3 4... is strictly incrasing w hav two cass to distinguish dpnding on th parity of th indics. To illustrat th tndncy a 3... a 14 ar listd blow up to svral digits: a 3 = 0.3... a 4 = 1.7... a 5 = 2.2... a 6 = 1.1... a 7 = 30.1... a = 362.9... a 9 = 711.9... a 10 = 11756.1... a 11 = 26250.9... a 12 = 244460.0... a 13 = 1.39 10 6 a 14 = 1.65 10 7. If is vn thn a < a +1 ( > 2) is quivalnt to b 2/(( 2)) < (!) 2/ /( 2) (( 1)!!) 1/( 2) ( + 1) (+1)/ 2 /( 2) and in th squl w may assum that 14. obtain < 2 By using induction w (2) b 2/(( 2)) ( 9). Indd supposing (2) and th rcursion for b +1 w hav to show ( ) 2 ( 2) ( ) ( + 1) (3) (!) 4 2 2 1 <. Assuming (3) not tru and applying! < ( ) +1 2 ( > 2) w obtain 2 1 22 4 2 4 ( + 1) 2 2 4 2 and ( 2.6 2 4 ( + 1)2 > 2 2 1 + 1 ) 2 2 4 which is fals for 14. Thrfor (3) and hnc (2) is provd for 14. On th othr hand ( ) 2 ( ) 1/2 < (!) 2/ < (( 1)!!) 1/( 2) and /( 2) ( + 1) (+1)/ > ( ) /( 2) + 1 ( ) /( 2) 14 > 15 ( ) 7/6 14 > 0.92 15
306 B. Brindza and Á. Pintér imply 2 / < 2.5 29.5 < ( ) 1/2 ( ) 2 0.92 2 7/6 ( 14) hnc a < a +1 is provd if is vn. Th rmaining cas ( is odd) is simpl. W gt ( ) 1/( 1) b a = a +1 = (2 +1 (!) 2 b ( + 1) (+1) (!!) 1 ) 1/( 1). On can obsrv that! >!! and < and thus Thorm 1 is provd. ( + 1) ( + 1) +1 < 2+1! ( + 1) +1 P r o o f o f T h o r m 2. Th xcptional cas ( m) = (3 3) is covrd by a rathr gnral rsult of [S1] (cf. [ST p. 122]). St A() = (1 3... ( 1)) 2 2 if is vn (1 3... ( 2)) 2 2 if is odd. As a mattr of fact w prov a littl mor. Namly th quation af (x) = bf m (y) in positiv intgrs x and y with aa() > b(m 1)! has only finitly many solutions. To guarant th conditions of Thorm A it is nough to show that (4) a min f (x) > b max f m(y). f (x)=0 f m (y)=0 Obviously b(m 1)! > b max f m(y). f m (y)=0 Sinc all th zros of f (x) ar ral also all zros of f (x) ar ral and by Roll s thorm thy altrnat with th zros of f (x). Elmntary calculus yilds ( ( a min f (x) > a min f 1 ( f 2) (x)=0 f 3 (... 2) f 2 3 ) ). 2
Irrducibility of som polynomials 307 Sinc ( a f 2j 1 ) (2( j) 1)!!(2j 1)!! = 2 2 a 1 3... (2j 1) 1 3 5... (2 (2j + 1)) = 2 a A() (j = 1... 1) (4) is provd. Rfrncs [BS] R. Balasubramanian and T. N. Shory On th quation f(x + 1)... f(x + ) = f(y + 1)... f(y + m) Indag. Math. (N.S.) 4 (1993) 257 267. [DLS] H. Davnport D. J. Lwis and A. Schinzl Equations of th form f(x) = g(y) Quart. J. Math. 12 (1961) 304 312. [MB] R. A. MacLod and I. Barrodal On qual products of conscutiv intgrs [SS] Canad. Math. Bull. 13 (1970) 255 259. N. Saradha and T. N. Shory Th quations (x+1)... (x+) = (y+1)... (y+ m) with m = 3 4 Indag. Math. (N.S.) 2 (1991) 49 510. [SST1] N. Saradha T. N. Shory and R. Tijdman On th quation x(x + 1)...... (x + 1) = y(y + d)... (y + (m 1)d) m = 1 2 Acta Arith. 71 (1995) 11 196. [SST2] On arithmtic progrssions with qual products ibid. 6 (1994) 9 100. [S1] A. Schinzl An improvmnt of Rung s thorm on diophantin quations Commnt. Pontific. Acad. Sci. 2 (1969) no. 20 1 9. [S2] Rducibility of polynomials of th form f(x) g(y) Colloq. Math. 1 (1967) [Sh] [ST] [Y] 213 21. T. N. Shory On a conjctur that a product of conscutiv positiv intgrs is nvr qual to a product of m conscutiv positiv intgrs xcpt for 9 10 = 6! and rlatd qustions in: Numbr Thory (Paris 1992 93) London Math. Soc. Lctur Not Sr. 215 Cambridg Univ. Prss Cambridg 1995 231 244. T. N. Shory and R. Tijdman Exponntial Diophantin Equations Cambridg Univ. Prss Cambridg 196. P. Z. Yuan On a spcial Diophantin quation a ( x n) = by r + c Publ. Math. Dbrcn 44 (1994) 137 143. Mathmatical Institut of Kossuth Lajos Univrsity P.O. Box 12 H-4010 Dbrcn Hungary E-mail: apintr@math.lt.hu Rcivd on 31.12.1996 and in rvisd form on 9.6.1997 (310)