Linear Forms in Logarithms and Applications

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.

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Applications of Linear Forms in Logarithms

Number Theory - Graduate Texts in Mathematics ◽

10.1007/978-0-387-49894-2_4 ◽

2008 ◽

pp. 411-440

Author(s):

Yann Bugeaud ◽

Guillaume Hanrot ◽

Maurice Mignotte

Keyword(s):

Linear Forms In Logarithms ◽

Linear Forms

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Mordell’s equation: a classical approach

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157015000182 ◽

2015 ◽

Vol 18 (1) ◽

pp. 633-646 ◽

Cited By ~ 4

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.

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The hyperelliptic equation over function fields

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100060497 ◽

1983 ◽

Vol 93 (2) ◽

pp. 219-230 ◽

Cited By ~ 28

Author(s):

R. C. Mason

Keyword(s):

Elliptic Equation ◽

Number Field ◽

Algebraic Number ◽

Algebraic Number Field ◽

Linear Forms In Logarithms ◽

Linear Forms ◽

Integer Coefficients ◽

Finite Set ◽

Integer Solutions ◽

Explicit Bounds

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

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