NUMBER OF REPRESENTATIONS OF INTEGERS BY BINARY FORMS

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1 NUMBER O REPRESENTATIONS O INTEGERS BY BINARY ORMS DIVYUM SHARMA AND N SARADHA Abstract We give improved upper bounds for the number of solutions of the Thue equation (x, y) = h where is an irreducible binary form of degree 3 Introduction Throughout this paper, let (x, y) = a 0 x r + a x r y + + a r y r be an irreducible binary form of degree r 3, with integer coefficients We assume without loss of generality that the content of, ie gcd(a 0,, a r ) is Let h be a nonzero integer In a seminal work in 909, Thue proved that the equation () (x, y) = h has only finitely many solutions in integers x and y or this purpose, he developed a method based on Diophantine approximation of algebraic numbers by rationals Since then, these equations are known as Thue equations Thue s method does not give any bound for the size of solutions, thus it is ineffective Nevertheless, it can be used to give bounds for the number of solutions Let N (r, h) denote the number of primitive solutions of (), ie, solutions (x, y) with gcd(x, y) = In 929, Siegel proposed that N (r, h) can be bounded by a function depending only on r and h, otherwise independent of, ie, there exists a positive number Z(r, h) depending on r and h such that N (r, h) Z(r, h) for any form of degree r In 983, Evertse [5] showed that Z(r, h) can be taken as 7 5((r 3)+) (r 3)(ω(h)+) 2000 Mathematics Subject Classification Primary D45; Secondary J68 Key words and phrases Binary forms, Thue-Siegel Principle, Approximation of algebraic numbers by rationals

2 2 DIVYUM SHARMA AND N SARADHA where ω(h) denotes the number of distinct prime factors of h closely related equation is (2) (x, y) = h If (x, y) is a solution of (2), then ( x, y) is also a solution of (2) Let N () (r, h) denote the number of primitive solutions of (2) with (x,y) and (-x,-y) identified as one solution Clearly, N (r, h) + N (r, h) = (r, h) Suppose 2N () (3) N () (r, h) Z() (r, h) Then N (r, h) + N (r, h) 2Z () (r, h) Thus we may take Z(r, h) = 2Z () (r, h) In 987, Bombieri and Schmidt [2] proved that there is an absolute constant c such that N () (r, h) c r +ω(h), for any form of degree r urther they showed that c = 25 if r is large In 99, Stewart [2] showed that (4) N (r, h) = 2800 ( + ) r +ω(g) 8ɛr for all r 3 with ɛ any positive real number and g is a divisor of h satisfying some conditions (see (7) below or Theorem of [2]) This is a refinement of the result of Bombieri and Schmidt In the above results, the method is based on counting large and small solutions for certain forms equivalent to and having large discriminant In [3], Zhang and Yuan obtained N () (r, h) 2r+ω(h) if D() 9 r(r ) and r 24 They gave similar bounds for 4 r 23 On the other hand, using linear forms in logarithms and geometry of numbers, Akhtari [] has shown that N () (r, h) (r 2)rω(h) if the discriminant of is larger than some effectively computable constant which depends only on r In the results of [],[2],and [3], the exponent of r is + ω(h) In fact, following [2], in these papers, it was enough to find an upper bound for N () (r, ) and then N () (r, h) rω(h) N () (r, ) Upper bounds for N () (r, h) were also considered by Győry in [6] and [7] One may easily see from the result of Győry in [6, Corollary 3] A

3 that if BINARY ORMS 3 N () + 8 (r, h) 25r + (r + 2)θ 4θ D() > r r (35 r h 2 ) 2(r ) θ for any θ with 0 < θ < In [7, Theorem G(ii)], by fixing θ = /3, he showed that N () (r, h) 32r + if D() (3 r h) 6(r ) We improve the results mentioned above from the papers [], [2], [6], [7] and [3] as follows Theorem We have where (5) c 0 = N () (r, h) c 0r +ω(h) { 20 i f r i f 4 r 22 or 3 r 3, the values of c 0 are given in Table urther N () (r, h) c 0 r+ω(h) i f D() p r(r ) 0 where (0, 29) i f r 24 (6) (c 0, p 0) = (0, 53) i f 8 r 23 (088, 53) i f 4 r 7 or 3 r 3, the values of (c 0, p 0) are given in Table Note (i) The value of c 0 in (5) corresponds to c = 25 in the result of Bombieri and Schmidt mentioned earlier Thus Theorem is explicit and gives a better estimate for all r 23 (ii) The value of c 0 in (6) is better than that of [] for r We do not use linear forms in logarithms and geometry of numbers as in [] or all values of r 4, the value of c 0 is better than those obtained in [3] (iii) We choose p 0 large to make c 0 small On the other hand, since it is known that c 0 = c 0 (p 0 + ), it is calculated by choosing that p 0 for which c 0 (p 0 + ) is small

4 4 DIVYUM SHARMA AND N SARADHA In 938, Erdős and Mahler [4] had shown that if has nonzero discriminant, h > c 2 and g a divisor of h with g > h 6/7 then N (r, h) c 3 r +ω(g), where c 2 and c 3 are positive numbers depending only on Stewart [2] improved this result as follows or any prime p and integers r 2, k and D 0, g 0, define Let T(r, k, p, D) = min { r ( ord p D k, min r 0 j r 2 (j + )(j + 2) + j j + 2 k)} Then (4) holds provided G(g, r, D()) = p T(r,ordpg,p,D()) (7) g +ɛ D() /r(r ) G(g, r, D()) h 2 r +ɛ, ɛ > 0 p g Remark Since r 3, the power of h in (7) is less than 6/7, thus sharpening the result of Erdős and Mahler Remark 2 Suppose g = h rom the definition of T, we get Hence T(r, ord p g, p, D()) ord pd() r(r ) + r 2 ord p g r G(g, r, D()) g r 2 r D() r(r ) h r 2 r D() r(r ) So (7) holds with g replaced by h Remark 3 It is well known that ω(h) has normal order log log h Suppose ψ(x, Y) denotes the Dickman function which counts the number of integers X having all its prime factors Y These are Y smooth numbers which are very well studied See [8] for a survey of smooth numbers The following estimate is due to Rankin, see [3] ψ(x, Y) X exp { log 3 Y log Y log X + log 2 Y + O ( )} log2 Y log 3 Y Taking X = h, we find that the number of integers not exceeding h and having very small prime factors are few in number Hence for a positive proportion of h, we may take g to be a prime satisfying (7) Then ω(g) = and we get Z(r, h) = 2800 ( + ) r 2 8ɛr

5 BINARY ORMS 5 c log h log log h There are values of h for which ω(h) is as large as with c an absolute constant while ω(g) = Hence the above estimate is much better We improve the result of Stewart in the following theorem Theorem 2 Suppose g is a divisor of h such that ( )) r ( (G(g, r, D())r h (8) D() max (, g r 2 g with µ = µ (r ), say Let c 0, c 0, p 0 be as in Theorem Then (i) N (r, h) 2c 0 r +ω(g) if µ = 283 (ii) N (r, h) 2c 0 r+ω(g) if D() p r(r ) and 0 { 3066 i f r 24 µ = 362 i f 4 r 23 and as in Table for 3 r 3 Remark 4 Assume G(g, r, D()) g 2/r On comparing the condition for D() in (8) with that of (7) due to Stewart, we find that (8) is better whenever 2 + ɛr µ Thus (8) is better for ɛ 28 if (i) holds and ɛ 045 for r 24; ɛ 6 for 4 r 23 if (ii) holds Similar remark holds for 4 r 3 by using Table Our method is based on the Thue -Siegel principle as enunciated in [2] and Diophantine approximation methods We divide the primitive solutions (x, y) according as 0 y < Y 0, Y 0 y < M() q and y M() q where Y 0 and q are chosen judiciously depending on r In fact, for the calculation of c 0 we find that Y 0 = 3 gives a better value and for computing c 0, Y 0 = 2 for r and Y 0 = for 4 r 0 yield better bounds This is a simple analogue of small, medium and large solutions considered by Mueller and Schmidt [] The parameter q is taken as 2 in all earlier works Here we find that it is more economical to take q smaller than 2 for large values of r or instance for r 24, q is taken as 54 for computing c 0 (see the proof of Theorem ) These choices result in the improved bounds given in Theorems and 2 2 Lemma on Discriminant Suppose γ,, γ r denote the roots of the equation (x, ) = 0 Denote by D() = a 2r 2 0 (γ i γ j ) 2 i<j ) µ

6 6 DIVYUM SHARMA AND N SARADHA and M() = a 0 r max(, γ i ), i= the discriminant and Mahler height of, respectively We begin with an elementary result which describes the change in the discriminant of a form when an element of GL(2, Z) acts on it ( ) a b Lemma 3 Let A = GL(2, Z) and let c d A (x, y) denote the form (ax + by, cx + dy) Then D( A ) = (deta) r(r ) D() Proof The coefficient of x r in A (x, y) is A (, 0) = (a, c) Therefore D( A ) = (a, c) 2r 2 i<j (β i β j ) 2, where β,, β r denote the roots of the equation A (x, ) = 0 Let i r Since A (β i, ) = 0, we have Hence which gives (aβ i + b, cβ i + d) = 0 aβ i + b cβ i + d = γ j for some j with j r β i = γ jd b a γ j c By changing the indices, if necessary, we may assume that or i j, we have β i = γ id b a γ i c, i r β i β j = γ id b a γ i c γ jd b a γ j c = (deta)(γ i γ j ) (a γ i c)(a γ j c) Observe that ((a, c)) r = (a γ i c)(a γ j c) i<j

7 BINARY ORMS 7 Therefore ( (deta)(γi γ j ) ) 2 D( A ) = (a, c) 2r 2 (a γ i c)(a γ j c) i<j = (deta) r(r ) a 2r 2 0 (γ i γ j ) 2 = (deta) r(r ) D() i<j 3 Equation (2) when h has a large divisor g Stewart [2] expanded the p adic technique of Bombieri and Schmidt [2] to reduce the problem of solving (2) to a set of equalities where the forms have large discriminant and h is reduced to h/g where g is a divisor of h satisfying some conditions The following lemma is an adaptation of ([2],Theorem ) Lemma 4 Let g be a divisor of h such that (9) g (r 2)(r ) D() G(g, r, D()) r(r ) ( ) µ h g Then, there is a set W of at most r ω(g) binary forms with the property that distinct primitive solutions (x, y) of (2) correspond to distinct triples (, x, y ) where is in W and (x, y ) is a pair of co-prime integers for which urther, if is in W, then (x, y ) = h g C( ) = and D( ) ( ) µ h g Proof We follow the arguments of the proof of Theorem in [2] Suppose (x, y) is a primitive solution of (2) Let p be a prime divisor of h and let k denote the highest power of p dividing h Then (x, y) 0 (mod p k ) Let p y Then (xy, ) 0 (mod p k ) Let Ω p be the completion of the algebraic closure of the field Q p of p-adic numbers We denote the p-adic value in Q p and its extension to Ω p by p Consider the ring R p of elements in Ω p whose p-adic value is Let s be the number of zeros of (z, ) in R p

8 8 DIVYUM SHARMA AND N SARADHA By Theorem 2 of [2], there is an integer t = t(k), 0 t s and integers b,, b t, u,, u t with 0 u i T = T(r, k, p, D()) such that xy b i ie, there is an integer A such that or i t, put (mod p k u i ) for some i x = p k u i A + b i y i (X, Y) = (p k u i X + b i Y, Y) By Theorem 2 of [2], p k divides C( i ) Since i (A, y) = (p k u i A + b i y, y) = (x, y) = h and k is the highest power of p dividing h, k is also the highest power of p dividing C( i ) Let q p be a prime dividing C( i ) Let P = p k u i and Q = b i Then i (X, Y) = (PX + QY, Y) So i (X, Y) = a 0 P r X r + (a 0 rp r Q + a P r )X r Y + + a r Y r Since q divides each of the coefficients and is co-prime to P, we obtain that q divides C(), which is Thus C( i ) = p k Put i (X, Y) = p k i (X, Y) Then C( i ) = Also ( ( ) ( ) p i (X, Y) = k/r 0 X ) i Therefore by Lemma 3, Also 0 p k/r = ( ( p k u i b i 0 Y ) ( p k/r 0 0 p k/r D( i ) = p 2k(r ) p (k u i)r(r ) D() ) ( X Y i (A, y) = p k i (x, y) = hp k f or i t Let now p y Then x is invertible modulo p k since gcd(x, y) = In this case, (, yx ) 0 (mod p k ) By Theorem 2 of [2] there exist integers w = w(k), b t+,, b w, u t+,, u w with 0 u i T = T(r, k, p, D()) such that ) ) yx b i (mod p k u i ) for some i with t + i w Let s be the number of zeros α of (, z) with α p < Then w t s Every non zero root of (, z) is the inverse of a non zero root of (z, )

9 BINARY ORMS 9 Hence s is the number of non zero roots γ of (z, ) with γ p > Therefore w t + s s + s = r We argue as before, with the roles of x and y interchanged to obtain y = p k u i A + b i x, and a form i (X, Y) = p k (X, b i X + p k u i Y) such that and D( i ) = p 2k(r ) p (k u i)r(r ) D() i (x, A ) = p k h f or t + i w By repeating this process for each prime factor of g, we get a set W of at most r ω(g) binary forms with the property that distinct primitive solutions (x, y) of (2) correspond to distinct triples (, x, y ) where is in W and (x, y ) is a pair of co-prime integers for which urther, Suppose g = p k pk l l Then (x, y ) = h g C( ) = D( ) = (p k u p k l u l l (p k pk l l )2r 2 ) r(r ) D() = gr(r ) g 2r 2 (p u p u l) r(r ) D() l g (r )(r 2) D() ( p g p T(r,ord pg,p,d()) ) r(r ) since u i T(r, ord p g, p, D()) for i l Thus D( ) ( g (r )(r 2) D() h G(g, r, D()) r(r ) g by (9) ) µ

10 0 DIVYUM SHARMA AND N SARADHA 4 orms with discriminant larger than a power of a prime By Lemma 4 and the discussions in the Introduction it is enough to consider forms satisfying (0) (x, y) = n with C() = and D() n µ where µ is as given in (9) Let N (2) (r, n) denote the number of primitive solutions of (0) We give an upper bound for N (2) (r, n) in terms of the number of solutions of forms having even larger discriminant Let p be a prime number and G an irreducible form of degree r satisfying () G(x, y) = n with C(G) = and D(G) p r(r ) n µ Let A SL(2, Z) Then G A has C(G A ) = and D(G A ) p r(r ) n µ Also G A (x, y) = n has the same number of solutions as G(x, y) = n Hence it is enough to consider () with G having smallest Mahler height among all forms SL(2, Z) equivalent to it Let N () (r, n; p) denote the maximum number of solutions of () for all forms G Lemma 5 We have N (2) (r, n) (p + )N() (r, n; p) urther for any form G with D(G) satisfying the condition in () we have (2) M(G) p r/2 n µ/(2r 2) r r/(2r 2) Proof Let (x, y) be a primitive solution of (0) Suppose ( ) ( ) p 0 0 A 0 = and A 0 j = for j p p j ( ) ( ) x x Then = A y j y for some j {0,,, p} and some integers x, y or, if x is divisible by p, we can take j = 0, x = x/p and y = y If x and p are co-prime, there exist integers a and b such that ax + bp = Now, y = x(ay) + p(by) = x(pq j) + p(by) for some integers q and j with j p = x( j) + p(xq + by)

11 BINARY ORMS Taking x = xq + by and y = x, we get gcd(x, y ) = and ( ) x A j y Since x and y satisfy (x, y) = n, x and y satisfy Aj (x, y ) = n Thus, for every primitive solution of (x, y) = n, there exists a primitive solution of Aj (x, y ) = n for some j with 0 j p So, if n j denotes the number of solutions of Aj (x, y) = n with 0 j p, we get N () (r, n) n 0 + n + + n p Also note that D( Aj ) p r(r ) Hence N () (r, n) (p + )N() (r, n; p) It is a well-known result of Mahler [0] that D() r r M() 2r 2 Therefore the Mahler height of such forms satisfies M() p r/2 n µ/(2r 2) r r/(2r 2) 5 Lemma of Lewis and Mahler ( x y The following lemma is a refinement, due to Stewart [2, Lemma 3], of an estimate of Lewis and Mahler [9] Lemma 6 Let G(x, y) be an irreducible form of degree r or any (x, y) with y 0, we have min α α x y 2r r (r )/2 (M(G)) r 2 G(x, y), D(G) /2 y r where the minimum on the left is over the roots of G(x, ) As an immediate consequence of the above lemma, we get the following corollary Corollary Let G be an irreducible form of degree r satisfying () Suppose µ 2 and p 3 Then (3) α 0 x α x := min M(G)r 2 y α y 2 y r ) =

12 2 DIVYUM SHARMA AND N SARADHA 6 Thue-Siegel Principle or the purpose of stating Thue-Siegel principle as given in ([2],p74), we introduce some notations Let t, τ be positive numbers such that Put and t < 2/r, 2 rt 2 < τ < t A = λ = 2 t τ t 2 (log M(G) + r ) 2 rt 2 2 Suppose that λ < r We say that a rational number x/y is a very good approximation to an algebraic number α of degree r if α x y < (4eA H(x, y)) λ where H(x, y) = max( x, y ) Lemma 7 If α is of degree r and x/y, x /y are two very good approximations to α, then where log(4e A ) + log H(x, y ) δ {log(4e A ) + log H(x, y)} or application we choose with 0 < a < b < Then δ = rt2 + τ 2 2 r t = 2/(r + a 2 ), τ = bt (4) λ = 2 ( b)t = 2(r + a2 ) b > 2r b, (5) A = ( log M(G) + r ) a 2 2 and δ = 2(b2 a 2 ) (r )(r + a 2 )

13 BINARY ORMS 3 7 Large Solutions In this section we estimate the number of primitive solutions (x, y) of () when y is large Lemma 8 Let G be a form satisfying () Suppose y M(G) q with q > Let B be a number satisfying (6) B = ( r ) 3 r 2r Then (7) x BM(G) y Proof By (3), α 0 x y M(G)r 2 = 2 y r p r 2 2M(G) (q )r+2 2M(G) 2 This implies that x y α 0 + 2M(G) M(G) + 2 2M(G) 2 by the definition of M(G) rom (2) and (6) we get B ( r ) 3 r 2r 2 2 2M(G) 3 Thus p r 2 x y M(G) + (B )M(G) = BM(G) This proves the lemma Let α i be a root of G(x, ) = 0 In the next lemma we count all those large primitive solutions (x, y) of () which are closest to α i Lemma 9 Let G be a form satisfying () or i r, set { I i = (x, y) : α i x y = min α α x } y and y M(G)q Let a, b, λ, δ and B be as given in sections 6 and 7 with r > λ Let (8) ν = log(4b) + (r/2a2 ) r log p r log r 2 2r 2 and η = (ν )λ 2 a 2 r λ

14 4 DIVYUM SHARMA AND N SARADHA Then [ ] log η log(q ) I i log(r ) log(r ) log (ν )r 2 a 2 δ((ν )λ 2) a 2 Proof Enumerate the primitive solutions in I i as (x, y ), (x 2, y 2 ), with y y 2 Put y j = M(G) +δ j Then + δj q > Hence δ j > 0 for j urther y j y j+ by (3) Thus, ie, Hence x j+ x j y j+ x j+ α i y j+ + α i x j M(G)r 2 y j It follows by induction that This also shows that y r j M(G) r 2 y j+ y j M(G) (+δ j)(r ) M(G) r 2 M(G) +δ j+ δ j+ (r )δ j δ j (r ) j δ f or j (9) δ j (r ) j (q ) for j Similarly, (20) δ k+l (r ) l δ k By the choice of ν and (2) we have y r j and by (5) and (7), M(G) ν 4Be r/2a2 (2) (4e A H(x j, y j )) λ (4e A BM(G) y j ) λ = M(G) By Corollary with α 0 = α i, we have α i x j M(G) 2 rδ j y j ( ) 2+ν+δj + λ a 2 Thus by (2), x j /y j is a very good approximation to α i if (22) rδ j + 2 (ν δ j + ) λ a 2

15 Let BINARY ORMS 5 [ ] log η log(q ) J := + log(r ) Then by (9), we have δ j η for j > J Thus by (22) and the definition of η, we find that x j /y j is a very good approximation to α i for j J + Claim The number of very good approximations to α i is at most (23) + log(r ) log (ν )r 2 a 2 δ((ν )λ 2) a 2 We prove the claim As seen above, x J+ /y J+ and x J+l /y J+l with l are very good approximations to α i Then by the Thue-Siegel principle, log(4e A ) + log H(x J+l, y J+l ) δ {log(4e A ) + log H(x J+, y J+ )} This implies that Since 4Be A M(G) ν+ a 2, log y J+l δ {log(4e A ) + log(bm(g)y J+ )} log y J+l δ { log ( M(G) ν+ a 2 By the definition of δ j s we get Thus by (20), B ) + log(bm(g)yj+ ) } ( + δ J+l ) log M(G) δ ( ν + a δ J+) log M(G) (r ) l δ J+l δ ( + ν + + 2) ( a2 δ + ν + + 2) a2 δ J+ δ J+ η = (ν a 2 )r 2 δ ( (ν a 2 )λ 2 ) Taking logarithm of both sides, we obtain (23) since the number of very good approximations is l Thus I i J + l which gives the assertion of the lemma

16 6 DIVYUM SHARMA AND N SARADHA 8 Small Solutions In this section we estimate the number of primitive solutions of () with Y 0 y < M(G) q where Y 0 is a positive integer Let x = (x, y) and set L i (x) = x α i y f or i r or x = (x, y) and x 0 = (x 0, y 0 ), let D(x, x 0 ) = xy 0 x 0 y We use the following estimate from ([2], Lemma 3 and (42)) Lemma 0 Let x = (x, y) be a solution of () Then there exists a number β i = β i (x) and an integer m = m(x) such that (24) ( m L i (x) β i ) y f or i r 2 Here, β,, β r are such that the form is equivalent to G Put { χ i = J(v, w) = n(v β w) (v β r w) x : G(x, y) = n, Y 0 y < M(G) q and L i (x) 2y Lemma Suppose x x with y ỹ belong to χ i Then (25) where β i = β i (x) and m = m(x) ỹ y 5Y max(, m β i ) Proof Since x D(x, x) = xỹ xy = x we have y ỹ = x α i y x α i ỹ } y ỹ, D(x, x) y L i ( x) + ỹ L i (x) y 2ỹ + ỹ L i(x) f or i r 2 + ỹ L i(x) By (24), ỹ 2 (( m β i ) ) y 2

17 Therefore, BINARY ORMS 7 ỹ y m βi 2( 2 ) y ( m βi Y 0 + 2) 2 This together with ỹ y shows that ỹ y max {, ( m βi Y 0 + 2)} 2 It is easy to see that the right hand side exceeds 5Y max{, m β i } which implies the assertion The following lemma is an immediate consequence of (24) Lemma 2 Suppose x is a solution of () with y Y 0 and L i (x) > /(2y) Then where β i = β i (x) and m = m(x) m β i Y 0 Proof This follows immediately from (24) Lemma 3 Let (r ) log p µ = log ( ) 5Y 0 +2 The number of primitive solutions of () with Y 0 y M(G) q does not exceed qr r + ( ) log 5Y log p 2r 2 log r provided the denominator in the expression above is positive Proof ix i r or each set χ i which is not empty, let x (i),, x(i) σ denote the elements of χ i ordered so that y y σ Put x(i) = x (i) σ Let χ be the set of solutions of () with Y 0 y, except x(),, x(r) By (25), x χ χ i ( 5Y max(, m(x) β i(x) ) ) σ i= y i+ y i y σ y M(G) q

18 8 DIVYUM SHARMA AND N SARADHA or x χ \ χ i, using Lemma 2, we get Therefore Note that x χ 5Y max(, m(x) β i(x) ( 5Y max(, m(x) β i(x) ) ) M(G) q i r max(, m(x) β i (x) ) = M(Ĵ) n where Ĵ = n(v (β m)w) (v (β r m)w) and Ĵ is equivalent to J and hence to G Thus we get max(, m(x) β i (x) ) M(G) n i r Hence ( ) r χ (M(G)/n) χ 5Y Thus Using (2), we get Since a+b c+d χ x χ i r M(G) qr 5Y max(, m(x) β i(x) ) qr log M(G) log M(G) log n r log ( ) 5Y 0 +2 χ max(a/c, b/d), χ qr ( ) log n+r log 5Y 0 +2 µ log n 2r r log p 2r 2 r log r qr min ( 2r 2, log µ Hence by the definition of µ we get qr χ log( 5Y0+2 ) 2 log p 2r 2 log r ( ) 5Y 0 +2 ) 2 log p 2r 2 log r So the number of solutions with Y 0 y < M(G) q is at most χ + r which gives the assertion of the lemma

19 BINARY ORMS 9 9 Parametric estimate for N () (r, n, p) Let G satisfy () Number of solutions of () with y Y 0, including (, 0) is at most (Y 0 )r + We now combine Lemmas 9 and 3 to get (26) N () (r, n, p) r(s + L) where S = Y 0 + r + q log ( 5Y 0 +2 ) 2 log p 2r 2 log r and [ ] log η log(q ) L = log(r ) log(r ) log (ν )r 2 a 2 δ((ν )λ 2), a 2 with λ, δ given by (4) and (5); ν, η given by (8) 0 Proof of Theorems and 2 Proof of Theorem As noted in the Introduction, we have N () (r, h) rω(h) N () (r, ) Thus it is enough to find an upper bound for N () (r, ) Note that (r, ) = N(2) (r, ) Hence by () and Lemma 4, N () (27) N () (r, ) { N () (r,, p) whenever D() p r(r ) (p + )N () (r,, p) otherwise or any given r, we choose a, b, p and Y 0 so that r > λ and the right hand side of (26) is as small as possible Let r 24 Take a = 5, b = 54, p = 29 and Y 0 = 2 Then η 22 Taking q = 54 > + η/(r ) we find that L 4 and S Thus the right hand side of (26) is at most 98608r which gives the assertion of Theorem, by (27) Let 4 r 23 In these cases take a = 4, b = 48, p = 53 and Y 0 = 2 urther take q = 204 for 8 r 23 and q = 2 for 4 r 7 Then the right hand side of (26) is at most 0r if 8 r 23 and 088r if 4 r 7 proving the assertion of Theorem Let 3 r 3 In these cases take q = 2 Also take Y 0 = 2 for r 3 & 3 and Y 0 = for 4 r 0 urther take a = 4, b = 48 if 9 r 3; a = 3, b = 36 if r = 6, 7, 8; a = 2, b = 24 if r = 4, 5 and a =, b = 5 if r = 3 The choice of p and the resulting c 0 is

20 20 DIVYUM SHARMA AND N SARADHA given in Table Note that the values of a, b, p are as in [3] but the values of c 0 obtained are always better than those given in [3] By choosing different values for a, b, p it is possible to get slightly improved bounds, but the improvement is not significant To obtain c 0, we choose Y 0 = 3; p = 7 for r 4 and p = 9 for r = 3 urther let (4, 48, 04) i f r 2 (4, 48, ) i f r = 0, (a, b, q) = (3, 36, ) i f 6 r 9 (2, 24, ) i f r = 4, 5 (, 5, ) i f r = 3 Then N (r, h) c 0 r +ω(h) where c 0 = 20, 236 if r 23, 4 r 22, respectively and for 3 r 3, c 0 is listed in Table This completes the proof of Theorem Note Let c 0 and p 0 be as in the statement of Theorem In the above proof, we showed that N () (r, h) c 0r if D() pr(r ) h µ 0 urther, as can be seen through Sections 5-0, these estimates hold not only for Thue equations but also for the Thue inequality (28) (x, y) h Since p 0 0 and µ 42(r ), our lower bound for D(G) and upper bound for the number of primitive solutions of (28) are both better than those in [7, Theorem G (ii)] Proof of Theorem 2 As noted in the Introduction, N (r, h) (r, h) and by Lemmas 4 and 5 we get 2N () N () (r, h) rω(g) N (2) (r, h/g) { r ω(g) (p + )N () (r, h/g; p) i f D() (h/g) µ r ω(g) N () (r, h/g; p) i f D() p r(r ) (h/g) µ Recall from Lemma 3 that µ = (r ) log p ( 5Y0 +2 log ) We make the same choices for a, b, p 0, Y 0 and q as in the proof of Theorem for each r It follows that { 2c0 r +ω(g) i f D() (h/g) µ (r ) N (r, h) 2c 0 r+ω(g) i f D() p r(r ) (h/g) µ (r ) 0

21 BINARY ORMS 2 with c 0, c 0 given by Theorem urther while calculating c 0, take µ = 283 for r 3 and when c 0 is calculated take µ = 3066, 362 for r 24, 4 r 23, respectively Also for 3 r 3, we record µ in Table This completes the proof of Theorem 2 Table r p 0 µ c 0 c Acknowledgement: We thank the referee for bringing to our attention the two papers [6] and [7] of K Győry

22 22 DIVYUM SHARMA AND N SARADHA References [] S Akhtari, Representations of unity by binary forms, Trans Amer Math Soc 364, 202, [2] E Bombieri and WM Schmidt, On Thue s equation, Invent Math 88, 987, 69-8 [3] N G de Bruijn, On the number of positive integers x and free of prime factors > y, Nederl Akad Wetensch Proc Ser A 54, 95, [4] P Erdős and K Mahler, On the number of integers which can be represented by a binary form, J London Math Soc 3, 938, [5] J H Evertse, Upper bounds for the number of solutions of diophantine equations, Math Centrum Amsterdam, 983, pp -27 [6] K Győry, Thue inequalities with a small number of primitive solutions, Periodica Math Hungar 42, 200, [7] K Győry, On the number of primitive solutions of Thue equations and Thue inequalities, Paul Erdos and his Mathematics I, Bolyai Soc MathStudies, 2002, [8] A Hildebrand and GTennenbaum, Integers without large prime factors, J Théor Nombres Bordeaux 5, 993, [9] D Lewis and K Mahler, Representation of integers by binary forms, Acta Arith 6, 96, [0] K Mahler, An inequality for the discriminant of a polynomial, Michigan Math J, 964, [] J Mueller and W M Schmidt, Thue s equation and a conjecture of Siegel, Acta Math 60 (988), [2] CL Stewart, On the number of solutions of polynomial congruences and Thue equations, J Amer Math Soc 4, 99, [3] P Yuan and Z Zhang, On the number of solutions of Thue equations, AIP Conf Proc 385, 20, 24-3 Address of the authors : School of Mathematics, Tata Institute of undamental Research, Homi Bhabha Road, Mumbai , INDIA s: divyum@mathtifrresin saradha@mathtifrresin