Linear Forms In Logarithms Research Articles

Diophantische Approximationen

Diophantische Approximationen

On the (non-)existence of m-cycles for generalized Syracuse sequences

This article generalizes a proof of Simons and de Weger [23] for the non-existence of (non trivial) m-cycles of the 3x + 1 problem to a proof for the (non)-existence of m-cycles for generalized Syracuse problems such as the 3x + q problem, the px + 1 problem, the px+q problem, the inverse Collatz problem, generalized Collatz problems and the Roelants problem. A lower bound for the cycle length is derived from a generalized lemma of Crandall [3]. For fixed m, an upper bound for the cycle length is found by theoretical approximation of the ratio between numbers in possible cycles and by applying results of Laurent a.o. [13] and Rhin [18] on linear forms in logarithms. For many generalized Syracuse sequences we theoretically and numerically prove the existence of all hypothetical non trivial m-cycles up to a given upper bound for m.

On the number of linear forms in logarithms

Let n be a positive integer. In this paper we estimate the size of the set of linear forms b 1 log a 1 + b 2 log a 2 + ⋯ + b n log a n , where | b i | ⩽ B i and 1 ⩽ a i ⩽ A i are integers, as A i , B i → ∞ .

Index of lattices and Hilbert polynomials

An upper bound for the index of a sublattice, which arises in relation to various versions of zero lemmas in the theory of linear forms in logarithms of algebraic numbers, in terms of the Hilbert polynomial is found. Simultaneously, a lower bound for the values of this polynomial is obtained.

Binomial Thue equations and polynomial powers

We explicitly solve a collection of binomial Thue equations with unknown degree and unknown -units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermat-type ternary equations, combining classical cyclotomy with Frey curve techniques.

Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.

Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue–Nagell equation x2 + D = yn, x, y integers, .

Exchanging the places p and [formula omitted] in the Leopoldt conjecture

The Leopoldt conjecture is concerned with the image of the global units in the local units at the primes dividing p. In the definition of the global units the infinite place is distinguished. Exchanging p and infinity in the formulation one gets a new conjecture. It predicts that certain vectors should be linearly independent over the reals whose components are arguments of conjugates of Weil numbers. Using Baker's result on linear forms in logarithms we prove part of this new conjecture in certain abelian situations.

Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over $\overline{\mathbb{Q}}$ , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

On the nonexistence of $2$-cycles for the $3x+1$ problem

This article generalizes a proof of Steiner for the nonexistence of 1-cycles for the 3x + 1 problem to a proof for the nonexistence of 2-cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of log(2) 3 shows that 2-cycles cannot exist.

Powers in recurrence sequences: Pell equations

In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the form x 2 - D y 2n = 1 in integers x, y and n > 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.

On px− qy= c and related three term exponential Diophantine equations with prime bases

Using a theorem on linear forms in logarithms, we show that the equation p x −2 y = p u −2 v has no solutions ( p, x, y, u, v) with x≠ u, where p is a positive prime and x, y, u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10 15. More generally, we obtain a similar result for p x − q y = p u − q v >0 where q is a positive prime, q≢1 mod 12 . We solve a question of Edgar showing there is at most one solution ( x, y) to p x − q y =2 h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions ( x, y) to | p x ± q y |= c and at most two solutions ( x, y, z) to p x ± q y ±2 z =0, for given positive primes p and q and integer c.

Uniformly counting points of bounded height

1. Introduction. In this paper we give some new uniform estimates for the cardinalities of certain sets involving algebraic numbers of bounded height. The estimates are nearly optimal with respect to the degree of the number eld. We mention some applications to problems about multiplicatively independent and dependent numbers in situations occurring in the recent theory of linear forms in logarithms associated with the name of Matveev. We also extend our counting estimates to algebraic vectors. Let K be a number eld of degree d = [K :Q] over the rational eld Q. We use the absolute height function dened for in K by

Criteria for irrationality of Euler’s constant

By modifying Beukers’ proof of Apéry’s theorem that ζ ( 3 ) \zeta (3) is irrational, we derive criteria for irrationality of Euler’s constant, γ \gamma . For n > 0 n>0 , we define a double integral I n I_n and a positive integer S n S_n , and prove that with d n = LCM ⁡ ( 1 , … , n ) d_n=\operatorname {LCM}(1,\dotsc ,n) the following are equivalent: 1. The fractional part of log ⁡ S n \log S_n is given by { log ⁡ S n } = d 2 n I n \{\log S_n\}=d_{2n}I_n for some n n . 2. The formula holds for all sufficiently large n n . 3. Euler’s constant is a rational number. A corollary is that if { log ⁡ S n } ≥ 2 − n \{\log S_n\}\ge 2^{-n} infinitely often, then γ \gamma is irrational. Indeed, if the inequality holds for a given n n (we present numerical evidence for 1 ≤ n ≤ 2500 ) 1\le n\le 2500) and γ \gamma is rational, then its denominator does not divide d 2 n ( 2 n n ) d_{2n}\binom {2n}{n} . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log ⁡ S n \log S_n . A by-product is a rapidly converging asymptotic formula for γ \gamma , used by P. Sebah to compute γ \gamma correct to 18063 decimals.

Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3x−2y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N⩾2, then the equation |(N+1)x−Ny|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.

On a few Diophantine equations, in particular, Fermat′s last theorem

This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat′s last theorem and its proof by Andrew Wiles using the modularity of elliptic curves, and we will exhibit other Diophantine equations which were solved à la Wiles. We will exhibit many families of Thue equations, for which Baker′s linear forms in logarithms and the knowledge of the unit groups of certain families of number fields prove useful for finding all the integral solutions. One of the most difficult conjecture in number theory, namely, the ABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

The Go polynomials of a graph

This paper introduces graph polynomials based on a concept from the game of Go. Suppose that, for each vertex of a graph, we either leave it uncoloured or choose a colour uniformly at random from a set of available colours, with the choices for the vertices being independent and identically distributed. We ask for the probability that the resulting partial assignment of colours has the following property: for every colour class, each component of the subgraph it induces has a vertex that is adjacent to an uncoloured vertex. In Go terms, we are requiring that every group is uncaptured. This definition leads to Go polynomials for a graph. Although these polynomials are based on properties that are less “local” in nature than those used to define more traditional graph polynomials such as the chromatic polynomial, we show that they satisfy recursive relations based on local modifications similar in spirit to the deletion–contraction relation for the chromatic polynomial. We then show that they are #P-hard to compute in general, using a result on linear forms in logarithms from transcendental number theory. We also briefly record some correlation inequalities.

Sur L'équation Diophantienne (x n −1)/(x −1)=y q , III

In this paper we study the diophantine equation of the title, which was first introduced by Nagell and Ljunggren during the first half of the twentieth century. We describe a method which allows us, on the one hand when n is fixed, to obtain an upper bound for q, and on the other hand when n and q are fixed, to obtain upper bounds for x and y which are far sharper than those derived from the theory of linear forms in logarithms. We also show how these bounds can be used even when they seem too large for a straightforward enumeration of the remaining possible values of x. By combining all these techniques, we are able to solve the equation in many cases, including the case when n has a prime divisor less than 13, or the case when n has a prime divisor which is less than or equal to 23 and distinct from q.2000 Mathematical Subject Classification: primary 11D41; secondary 11J86, 11Y50.

Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif

We obtain some new results in the theory of linear forms in logarithms on a commutative algebraic group, defined over Q . We generalize recent progress of S. David and N. Hirata [1]. In particular, we achieve an optimal linear independence measure in the height of the linear form and a measure more precise than Hirata's [4] for parameters related to the logarithms. The proof rests on Baker's method and a new argument of arithmetical nature.

On the exponential Diophantine equation $a^x + l^y = c^z$

Using the theory of linear forms in logarithms and a result due to Adachi, we show that the title equation has only the trivial positive integral solution under some conditions.