Shabnam Akhtari
We revisit a work by R. Okazaki and prove that for every cubic binary form F(x, y) with large enough discriminant, the Thue equation |F(x, y)| = 1 has at most 7 solutions in integers x and y.
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In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.
To each non totally real cubic extension $K$ of $\Q$ and to each generator $α$ of the cubic field $K$, we attach a family of cubic Thue equations, indexed by the units of $K$, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.
We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.
Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial solutions for $n\neq 1,3$.
We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.
In this paper we completely solve the family of parametrised Thue equations \[ X(X-F_n Y)(X-2^n Y)-Y^3=\pm 1, \] where $F_n$ is the $n$-th Fibonacci number. In particular, for any integer $n\geq 3$ the Thue equation has only the trivial solutions $(\pm 1,0), (0,\mp 1), \mp(F_n,1), \mp(2^n,1)$.
Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=λ\] where $λ>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $λ$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389…
We study the field isomorphism problem of cubic generic polynomial $X^3+sX+s$ over the field of rational numbers with the specialization of the parameter $s$ to nonzero rational integers $m$ via primitive solutions to the family of cubic Thue equations $x^3-2mx^2y-9mxy^2-m(2m+27)y^3=λ$ where $λ^2$ is a divisor of $m^3(4m+27)^5$.
Let $(A_n)_{n\in \mathbb{N}}, (B_n)_{n\in \mathbb{N}} \in \mathbb{Z}^{\mathbb{N}}$ be two linear-recurrent sequences that meet a dominant root condition and a few more technical requirements. We show that the split family of Thue equations \[ |X(X-A_n Y)(X-B_n Y) - Y^3| = 1 \] has but the trivial solutions $\pm\{ (0,1), (1,0), (A_n,1), (B_n,1) \}$, if the parameter $n$ is larger than some effectively computable constant.
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(λ)$ of D. Shanks, $λ$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $λ$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the …
For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.
In the first part we construct algorithms which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we u…
Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$ such that $h(x,y)$ is an $S$-unit. In the present paper, we reverse a well known argument, which seems to go back to Shafarevich, and use the modularity of elliptic curves over $\mathbb{Q}$ to give upper bounds for the number of solutions of such a Thue-Mahler equation. In addition, our methods gives an effective method for determining all solutions, and we …
Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the "small" solutions of these equations. Under "small" solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue e…
A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.
We consider Thue equations of the form $ax^k+by^k = 1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $ax^k+by^k+cz^k = 0$ of degree at least six.
We consider binomial Thue equations of type $x^n-my^n=\pm 1$ in $x,y\in Z$. Optimizing the method of Peth\H o we perform an extensive calculation by a high performance computer to determine all solutions with $\max(|x|,|y|)<10^{500}$ of binomial Thue equations for $m<10^7$ for exponents $n=3,4,5,7,11,13,17,19,23,29$.
The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle forms whose discriminant is small in absolute value. We then apply our results to bound the number of integral points on a certain family of elliptic curves.
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $α$ such that $K(α)$ is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.
In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers when $d \geq 3$. Moreover, when such a solution exists, it is of the form $(u,\sqrt{v})$ where $(u,v)$ is the fundamental solution of $X^{2}+1=dY^{2}$.
We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve $ϕ(x, y)$ that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki by giving special treatments to forms with respect to their signature.
For every $a\geq1$ we give a recursion algorithm of building of set of solutions of equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x),$ where $\{t(n)\}$ is Thue-Morse sequence. We pose an open problem and two conjectures.
In our recent paper we gave an efficient algorithm to calculate "small" solutions of relative Thue equations (where "small" means an upper bound of type $10^{500}$ for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.
In this paper, we prove that a Thue equation F(x,y) = h, where h is an integer and F is a polynomial of degree n with integer coefficients and without repeated roots, has at most 2n^3 - 2n - 3 solutions provided that the Mordell-Weil rank of the Jacboian of the corresponding curve is less than (n-1)(n-2)/2. The proof uses the method of Coleman-Chabauty, extended here to apply to arbitrary regular models of curves, along with an explicit construction of a portion of a regular model for the curve corresponding to the Thue equation.
Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every su…
We consider the parametric family of sextic Thue equations \[ x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=λ\] where $m\in\mathbb{Z}$ is an integer and $λ$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.
We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducin…
Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ be a binary form such that \[ F(x , y) =(αx + βy)^r -(γx + δy)^r, \] where $α$, $β$, $γ$ and $δ$ are algebraic constants with $αδ-βγ\neq 0$. We establish upper bounds for the number of primitive solutions to the Thue inequality $0<|F(x, y)| \leq h$, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.
Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions, on the other hand to compute all solutions (i.e. to prove the non-existence of large solutions) is a time consuming procedure. Therefore it is very practical to have a fast algorithm to calculate the "small" solutions, especially if "small" means less than e.g. $10^{100}$. Such an algorithm was constructed by A.Peth\H o in 1987 based on continued fractions. In the present paper we construct a similar type of fast algorithm to calculate "small" solutio…
In this semi-expository article we solve the diophantine equation $m^5+(4 \cdot 5^4 b^4)mn^4 - n^5=1$ for all integers $b \neq 0$. This gives an example of a family of quintic Thue equations that can be solved completely by using nothing more than Skolem's $p$-adic method. We also give a general introduction to Skolem's method from a modern perspective.
We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-λY)+Y^4=ξ,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter $λ\in\mathbb{C}[T]$ is some non-constant polynomial and $0\neqξ\in\mathbb{C}$. It is a function field analogue of the family solved by Mignotte, Pethő and Roth in the integer case. A feature of our proof is that we avoid the use of height bounds by considering a smaller relevant ring for which we can determine the units more easily. Because of this, the proof is short and the arguments are…
Thue equations that simultaneously fail the Hasse principle
改进构造，得到正比例图厄方程，对小于正整数m的h，其F(x; y)=h同时不满足整数哈塞原理，F为固定次数>2的二元形式且按系数绝对值最大者排序。
Complete Resolution Of A Family Of Twisted Thue Equations
本文求解了Lévesque与Waldschmidt提出的一类扭曲Thue方程，找出其所有整数解，解答了相关公开问题。
For any fixed nonzero integer $h$, we show that a positive proportion of integral binary quartic forms $F$ do locally everywhere represent $h$, but do not globally represent $h$. We order classes of integral binary quartic forms by the two generators of their ring of $\textrm{GL}_{2}(\mathbb{Z})$-invariants, classically denoted by $I$ and $J$.
Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_1, \dots, γ_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( γ_1^{t_1}\cdots γ_s^{t_s} \right) = K$, all of which are effectively computable.
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
On split families of Thue equations with linear recurrence sequences as factors
研究以线性递推序列为参数函数的 Thue 方程分裂族，证实了在参数大于某可计算常数时存在显式解集的结论对特定递推序列族同样成立。
Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family
针对$\mathbb{C}(T)$上的一类简单四次Thue方程族，本文应用ABC定理求出了该族方程在$\mathbb{C}[T] \times \mathbb{C}[T]$上的全部解，完成了该族方程的完全求解。
We will use Thue-Siegel method, based on Padé approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, for a certain family of irreducible quartic binary forms.
We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler's generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue--Mahler for degree $3$ implies the theorem of Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line $¶^1…
Let $F$ be an irreducible binary form attached to a number field $K$ of degree $\geq 3$. Let $ε\not\in \{-1, 1\}$ be a totally real unit of $K$. By twisting $F$ with the powers $ε^a$ of $ε$, ($a\in{\mathbf Z}$), we obtain an infinite family $F_a$ of binary forms. Let $m\in{\mathbf Z}$. We give an effective bound for $\max\{|a|, \log|x|, \log|y|\}$ when $a,x,y$ are rational integers satisfying $F_a(x,y)=m$ with $xy\not=0$.
Let $α$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(α)$. For any unit $ε$ of $K$ such that ${\mathbf Q}(αε)=K$, we consider the irreducible polynomial $f_ε(X)\in{\mathbf Z}[X]$ such that $f_ε(αε)=0$. Let $F_ε(X,Y)\ = Y^df_ε(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,ε)$ of the diophantine equation $|F_ε(x,y)|\leq m$.
Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $μ$, $α_1,α_2,α_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by $\OS^\times$ the group of $S$--units. Then the set of equivalence classes (namely, up to multiplication by $S$--units) of the solutions $(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon)\in\OS^3\times(\OS^\times)^4$ of the diophantine equation $$ (X-α_1 E_1 Y) (X-α_2E_2 Y) (X-α_3E_3 Y)Z=μE, $$ satisfying $\Card\{α_1\varepsilon_1,α_2\varepsilon_2,α_3\varepsilon_3\}= 3$, …
Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.
Power-saving bounds for Thue--Mahler and Mordell equations
该研究针对三次Thue--Mahler方程证明了对regulator具节能依赖的有效界，将Mordell方程整数解的Stark界指数降至1/2+ε，改进了相关分离下界并在Frey高度猜想的导体方面取得节能进展。
Let $K$ be a field of char $K\neq 2$. For $a\in K$, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial $X^4-aX^3-6X^2+aX+1$ over $K$ as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field $K$, we see that the polynomial gives the same splitting field over $K$ for infinitely many values $a$ of $K$. We also see by Siegel's theorem for curves of genus zero that only finitely many algebraic integers $a\in\mathcal{O}_K$ in a number field $K$ may give the same splitting field. By appl…
This paper announces the discovery of an isoperimetric inequality for the area of plane regions defined by binary forms. This result has been applied subsequently in the enumeration of solutions to the Thue inequality and, given its fundamental nature, may find application in other areas as well.
In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible (over $\mathbb{Z}$) binary form with degree $n \geqslant 3$ and exactly three nonzero summands. In particular, we improve Emery Thomas' explicit upper bounds on the number of solutions to this equation. For instance, when $n \geqslant 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work, where he …
Let $α$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(α)$. When $\varepsilon$ is a unit of $K$ such that $\Q(α\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X) \in \Z[X]$ such that $f_\varepsilon(α\varepsilon)=0$. Let $F_\varepsilon(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\varepsilon}(X) $ under the condition $F_{\varepsilon}(X,1)=f_{\varepsilon}(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\varepsilon)$ of the diophantine inequation $|F_\va…
Efficient resolution of Thue-Mahler equations
本文提出一种仅需在添加单根的数域中计算的新算法，并引入格筛技术，高效求解高次Thue-Mahler方程，应用确定49对满足$P(X^4-2Y^4)\le100$的互素非负整数对。
We consider the Diophantine equation X^n - 1 = B.Z^n, where B in Z is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, phi(rad (B)), has a common divisor with the exponent n, or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell-Ljunggren equation: (X^n-1)/(X-1) = n^e.Y^n; e = 0 or 1. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution…
Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
Padé approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of Siegel's lemma to sharpen the estimates of Padé-type approximations, or by finding completely explicit expressions for the yet unknown 'twin type' Hermite-Padé approximations. The appropriate homogeneous matrix equation representing both methods has an $M \times (L+1)$ coefficient matrix, where $M \le L$. The homogeneous solution vectors of this matrix equation…
Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor.
Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $σ_1,σ_2,\dots,σ_d$ the embeddings of $K$ into $\mathbb{C}$, $α$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-σ_i(α) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-σ_i(α\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effect…
Mahler's work on Diophantine equations and subsequent developments
本文讨论Mahler在丢番图逼近及其在丢番图方程中应用的工作，并深入探讨Thue-Mahler方程解的个数与有效有限性等后续发展。
A Proof of the Thue-Siegel Theorem about the Approximation of Algebraic Numbers for Binomial Equations
本文通过推广二项级数的连分数展开并构造代数逼近函数及相应积分，在不使用鸽巢原理的情况下，为二项方程代数数证明了带 Siegel 指数的 Thue 定理。
We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are independent of $m$, when $m$ is smaller than $|D|^{\frac{1}{4(n-1)}}$. We also consider the Thue equation $|F(x , y)| = m$ and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer $m$, when $ m < |D|^{\frac{1}{2(n-1)}}$.
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