We show how to determine the $k$-th bit of Chaitin's algorithmically random real number $Ω$ by solving $k$ instances of the halting problem. From this we then reduce the problem of determining the $k$-th bit of $Ω$ to determining whether a certain Diophantine equation with two parameters, $k$ and $N$, has solutions for an odd or an even number of values of $N$. We also demonstrate two further examples of $Ω$ in number theory: an exponential Diophantine equation with a parameter $k$ which has an odd number of solutions iff the $k$-th bit of $Ω$ is 1, and a polynomial of positive integer variables and a parameter $k$ that takes on an odd number of positive values iff the $k$-th bit of $Ω$ is 1.
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 $X^{2}+1=dY^{2}$.
In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for n=3, a 5-parameter of solutions; likewise for n=4. For n=5, a 7-parameter family of solutions; and likewise for n=6. See Theorems 2 through 6 respectively. The second goal of this paper, is determining all the positive integer solutions of xyz=w^2. This is done in Theorem7; the solution set is described in terms of six independent parameters. Finally, in Theore…
We solve a family of quadratic Diophantine equations associated to a simple kind of games. We show that the ternary case, in many ways, is the most interesting and the least arbitrary member of the family.
In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operati…
Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.
We show the insolvability of the Diophantine equation $ax^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1 \pmod {12}$ and $b=2^da-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we investigate the more general family with $b=2^da-3^r$, where $r$ is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over $\mathbb{Q}$. We conclude by providing some numerical evidence corroborating the main results.
We consider a one-parameter family of third order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd. Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a cer…
In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p congruent to 7 modulo 8. In addition to the above, assume that one of the following holds: Either (i) n^2-p>0 and the positive integer is a prime, Or (ii) n^2-p<0 and the positive integer N=-m=-(n^2-p) is a prime. Then the diophantine equation x^4 +2nx^2y^2+my^4=z^2 has no positive integer solutions. The method of proof is elementary in that it only uses co…
Let $\F$ be a family of $r$ integral forms of degree $k\geq 2$ and $\LL=(l_1,\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\x=(x_1,...,x_n)$. We study the number of solutions $\x\in[1,N]^n$ to the diophantine system $\F(\x)=\vv$ under the restriction that $l_i(\x)$ has a bounded number of prime factors for each $1\leq i\leq m$. We show that the system $\F$ have the expected number of such "almost prime" solutions under similar conditions as was established for existence of integer solutions by Birch.
In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.
One of Legendre's theorems on the Diophantine equation $ax^2+by^2+cz^2=0$ provides necessary and sufficient conditions on the existence of nonzero rational solutions of this equation, which helps determine the existence of rational points on a conic. In this paper, we provide a new proof of this famous theorem using Hasse invariant and Jacobi symbol from the theory of quadratic forms.
We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral point.
In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.
We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.
This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $\frac{(m-1)(m+n-2)}{2} $ or $\frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd. We find the solution of a diophantine system of equations by using the solution of the diophantine equation. We generalized all the results of the paper [5].
In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea of a simple chain to an $n$-chain in which $n$ is a positive integer and if a pair $(u_{i-1}, u_{i})$ has a certain property, then the $n$th next pair $(u_{\pm n+i-1}, u_{\pm n+i})$ also has the same property. In this case, we call $(u_{i-1}, u_{i}, u_{i+1})$ and $(u_{\pm n+i-1}, u_{\pm n+i}, u_{\pm n+i+1})$ matching triples. We use $4$-chains to study a fa…
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite sets.
Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in …
In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.
We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) = K_{n+1}(x_0,\ldots,x_n)K_{n+1}(x_1,\ldots, x_{n+1})$. We show that with certain restrictions on $P(x)$ the set of its solutions has a rich structure. In particular, we provide several ways of generating new solutions from the existing ones. In the last section we discuss the relation between the solutions of the above Diophantine equation for arbitrary values of $n$ an…
In this paper we focus on finding all the factorials expressible as a product of a fixed number of $2k$-nacci numbers with $k \geq 2$. We derive the 2-adic valuation of the $2k$-nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with $p$-regular sequences.
For K \subseteq C, let B_n(K)={(x_1,...,x_n) \in K^n: for each y_1,...,y_n \in K the conjunction (\forall i \in {1,...,n} (x_i=1 => y_i=1)) AND (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k => y_i+y_j=y_k)) AND (\forall i,j,k \in {1,...,n} (x_i*x_j=x_k => y_i*y_j=y_k)) implies that x_1=y_1}. We claim that there is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer n>=m(f), and returns a tuple (x_1,...,x_n) \in B_n(Z) with x_1=f(n). We compute an integer tuple (x_1,...,x_{20}) for which the stat…
We study the existence of rational points on modular curves of $\cal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of $\cal{D}$-elliptic sheaves, and constructing curves violating the Hasse principle.
In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for odd prime $p$ and guarantee the existence of a $k$-Diophantine m-tuple provided $p$ is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in $\mathbb{F}_p$ as well as an asymptotic formula for the number of $k$-Diophantine $k$-tuples.
Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effect…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids are described by a system of four Diophantine equation possessing a natural $S_3$ symmetry. Recently these equations were factorized with respect to this $S_3$ symmetry and the factor equations were derived. In the present paper the factor equations are transformed to $E$-form and then reduced to a single biquadratic equation.
We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.
Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation $$cx^2+p^{2m}=4y^n,~~x,y\geq 1, m\geq 0, n\geq 3, \gcd(x,y)=1, \gcd(n,2h(-c))=1,$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.
We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain Diophantine equation of the order 12. In this paper a structural theorem for the solutions of this Diophantine equation is proved.
The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert's 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smalles…
We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine …
We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $ω_2(ξ)$, $\widehatω_2(ξ)$, $λ_2(ξ)$, $\widehatλ_2(ξ)$ associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new info…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The second cuboid conjecture is one of the three propositions suggested as intermediate stages in proving the non-existence of perfect Euler cuboids. It is associated with a certain polynomial Diophantine equation of the order 10. In this paper a structural theorem for the solutions of this Diophantine equation is proved and some examples of its application are considered.
Let $\ell, m, r$ be fixed positive integers such that $2\nmid \ell$, $3\nmid \ell m$, $\ell>r$ and $3\mid r$. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if $\min\{r\ell m^2-1,(\ell-r)\ell m^2+1\}>30$, then the equation $(r\ell m^2-1)^x+((\ell -r)\ell m^2+1)^y=(\ell m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2…
In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrtε)/\mathbb{Q}(\sqrt{d})$ (where $ε$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb…
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.
Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. A conjecture of Hall attempts to ameliorate this by bounding the size of integral solutions simply in terms of the coefficients of the defining equation. It turns out that a simi…
A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessar…
In this paper, we construct a family of elliptic curves with rank $\geq 5$. To do this, we use the Heron formula for a triple $(A^2, B^2, C^2)$ which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers $A$, $B$, and $C$, along with the extra parameter $D$ satisfy the quartic Diophantine equation $A^4+D^4=2(B^4+D^4)$.
We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a res…
We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\to\infty} |n|<na - x> <nb - y> = 0, where <-> denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit cl…
Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A > 8 B^3$, then the equation $(*) \quad (A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$. Combining the above conclusion with some existing results, we can deduce that if $A >8 B^3$ and $B \equiv 2 \bmod 4$, then (*) has o…
This paper deals with three consequences of the existence of Arithmetic Teichmuller spaces of arXiv:2106.11452. Let $\mathscr{X}_{F,\mathbb{Q}_p}$ (resp. $B=B_{\mathbb{Q}_p}$) be the complete Fargues-Fontaine curve (resp. the ring) constructed by Fargues-Fontaine with the datum $F={\mathbb{C}_p^\flat}$ (the tilt of $\mathbb{C}_p$), $E=\mathbb{Q}_p$. Fix an odd prime $\ell$, let $\ell^*=\frac{\ell-1}{2}$. The construction (§7) of an uncountable subset $Σ_{F}\subset \mathscr{X}_{F,\mathbb{Q}_p}^{\ell^*}$ with a simultaneous valuation scaling property (Theorem 7.8.1), Galois action and other symm…
Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $δ(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lovász local lemma to es…
In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+αX_{2}+α^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(α)$, $α$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if $\op{deg}f=4$ and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirat…
In this paper, we consider the Diophantine equation $λ_1U_{n_1}+\ldots+λ_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots,p_s$ are fixed, distinct prime numbers; $λ_1,\dots,λ_k$ are strictly positive integers; and $n_1,\dots,n_k,z_1,\dots,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots,n_k,z_1,\dots,z_s)$. In our proof, we use lower bounds for linear forms in log…
Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $ξ$ that is either transcendental or algebraic of sufficiently large degree…
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.
