scholarly journals Computing the effectively computable bound in Baker's inequality for linear forms in logarithms, and: Multiplicative relations in number fields: Corrigenda and addenda

1977 ◽  
Vol 17 (1) ◽  
pp. 151-155
Author(s):  
A. J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Number Fields ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Computable Bound

We indicate a number of qualifications and amendments that are necessary so as to correct the statements and proofs of the theorems in our papers “Computing the effectively computable bound in Baker's inequality for linear forms in logarithms”, 15 (1976), 33–57, and its sequel “Multiplicative relations in number fields”, 16 (1977), 83–98, and remark on recent observations that would yield yet sharper results.

scholarly journals On a few Diophantine equations, in particular, Fermat's last theorem

2003 ◽  
Vol 2003 (71) ◽  
pp. 4473-4500
Author(s):  
C. Levesque
Keyword(s):  
Diophantine Equations ◽  
Number Fields ◽  
Linear Forms In Logarithms ◽  
Open Problems ◽  
Abc Conjecture ◽  
Linear Forms ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
The Subject ◽  
Integral Solutions

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 solvedà laWiles. 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, theABC conjecture, will also be described. We will conclude by explaining in elementary terms the notion of modularity of an elliptic curve.

The S-unit equation over function fields

1984 ◽  
Vol 95 (1) ◽  
pp. 3-4 ◽  
Author(s):  
Joseph H. Silverman
Keyword(s):  
Algebraic Geometry ◽  
Diophantine Equations ◽  
Function Fields ◽  
Number Fields ◽  
Upper Bounds ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Image Position ◽  
The Given ◽  
Integral Solutions

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equationin S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

1976 ◽  
Vol 15 (1) ◽  
pp. 33-57 ◽  
Author(s):  
A.J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
Computable Bound ◽  
Image Position ◽  
Detailed Computation

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

2020 ◽  
Vol 102 (3) ◽  
pp. 479-489
Author(s):  
XIANG GAO ◽  
SHENGYOU WEN
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Transcendental Number ◽  
Linear Forms In Logarithms ◽  
Multiplicative Semigroup ◽  
Linear Forms ◽  
Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Applications of Linear Forms in Logarithms

2008 ◽  
pp. 411-440
Author(s):  
Yann Bugeaud ◽  
Guillaume Hanrot ◽  
Maurice Mignotte
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms

scholarly journals Mordell’s equation: a classical approach

2015 ◽  
Vol 18 (1) ◽  
pp. 633-646 ◽  
Author(s):  
Michael A. Bennett ◽  
Amir Ghadermarzi
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Classical Approach ◽  
Linear Forms In Logarithms ◽  
Lattice Basis Reduction ◽  
Linear Forms ◽  
Basis Reduction

We solve the Diophantine equation$Y^{2}=X^{3}+k$for all nonzero integers$k$with$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

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