scholarly journals A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations

2008 ◽  
Vol 60 (3) ◽  
pp. 491-519 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek
Keyword(s):  
Lower Bounds ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Arbitrary Degree ◽  
Famous Theorem ◽  
Linear Forms ◽  
Positive Integers ◽  
Level Lowering

AbstractWe solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation5uxn − 2r3s yn = ±1,in non-zero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

On Some Exponential Equations of S. S. Pillai

2001 ◽  
Vol 53 (5) ◽  
pp. 897-922 ◽  
Author(s):  
Michael A. Bennett
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Prior Work ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Sharp Result ◽  
Exponential Equations ◽  
Positive Integers

AbstractIn this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ax – by = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

INTEGRAL POINTS ON CONGRUENT NUMBER CURVES

2013 ◽  
Vol 09 (06) ◽  
pp. 1619-1640 ◽  
Author(s):  
MICHAEL A. BENNETT
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Precise Description ◽  
Integral Points ◽  
Linear Forms ◽  
Integer Points ◽  
Congruent Number

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 - N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of (non-trivial) computation.

scholarly journals On explicit estimates for linear forms in the values of a class of E-functions

1982 ◽  
Vol 25 (1) ◽  
pp. 37-69 ◽  
Author(s):  
Xu Guangshan ◽  
Wang Lianxiang
Keyword(s):  
Differential Equations ◽  
Lower Bounds ◽  
Linear Differential Equations ◽  
Linear Forms ◽  
Linear Differential

We apply methods of Mahler to obtain explicit lower bounds for certain combinations of E-functions satisfying systems of linear differential equations as studied by Makarov. Our results sharpen and generalise earlier results of Mahler, Shidlovskii, and Väänänen.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

The paucity problem for simultaneous quadratic and biquadratic equations

1999 ◽  
Vol 126 (2) ◽  
pp. 209-221 ◽  
Author(s):  
W. Y. TSUI ◽  
T. D. WOOLEY
Keyword(s):  
Lower Bounds ◽  
Diophantine Equations ◽  
Upper And Lower Bounds ◽  
Current Interest ◽  
Sieve Methods ◽  
Number Of Solutions ◽  
Simultaneous Diophantine Equations ◽  
History Of

The problem of constructing non-diagonal solutions to systems of symmetric diagonal equations has attracted intense investigation for centuries (see [5, 6] for a history of such problems) and remains a topic of current interest (see, for example, [2–4]). In contrast, the problem of bounding the number of such non-diagonal solutions has commanded attention only comparatively recently, the first non-trivial estimates having been obtained around thirty years ago through the sieve methods applied by Hooley [10, 11] and Greaves [7] in their investigations concerning sums of two kth powers. As a further contribution to the problem of establishing the paucity of non-diagonal solutions in certain systems of diagonal diophantine equations, in this paper we bound the number of non-diagonal solutions of a system of simultaneous quadratic and biquadratic equations. Let S(P) denote the number of solutions of the simultaneous diophantine equationsformula herewith 0[les ]xi, yi[les ]P(1[les ]i[les ]3), and let T(P) denote the corresponding number of solutions with (x1, x2, x3) a permutation of (y1, y2, y3). In Section 4 below we establish the upper and lower bounds for S(P)−T(P) contained in the following theorem.

scholarly journals Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1047
Author(s):  
Pavel Trojovský ◽  
Štěpán Hubálovský
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Recursive Relation ◽  
Initial Values ◽  
Lucas Sequence ◽  
Diophantine Problems ◽  
Positive Integers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

scholarly journals Explicit lower bounds for linear forms

2016 ◽  
Vol 85 (302) ◽  
pp. 2995-3008
Author(s):  
Kalle Leppälä
Keyword(s):  
Lower Bounds ◽  
Linear Forms

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

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.

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