Solving a specific Thue-Mahler equation

Tzanakis, Nikos; Weger, de Bmm Benne

doi:10.1090/s0025-5718-1991-1094961-0

Cited by 16 publications

(16 citation statements)

References 8 publications

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“…Next suppose that inequality (46) (and hence also inequality (47)) fails to hold. In this case, we will apply lower bounds for linear forms in two complex logarithms.…”

Section: The Modular Approach To Diophantine Equations: Some Backgroundmentioning

confidence: 99%

Differences between perfect powers : prime power gaps

Bennett¹,

Siksek²

2021

Preprint

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We develop machinery to explicitly determine, in many instances, when the difference x 2 − y n is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and non-archimedean, lattice basis reduction, methods for solving Thue-Mahler and S-unit equations, and the Primitive Divisor Theorem of Bilu, Hanrot and Voutier) and classical Algebraic Number Theory, with results derived from the modularity of Galois representations attached to Frey-Hellegoaurch elliptic curves. By way of example, we completely solve the equationx 2 + q α = y n , where 2 ≤ q < 100 is prime, and x, y, α and n are integers with n ≥ 3 and gcd(x, y) = 1.

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“…Next suppose that inequality (46) (and hence also inequality (47)) fails to hold. In this case, we will apply lower bounds for linear forms in two complex logarithms.…”

Section: The Modular Approach To Diophantine Equations: Some Backgroundmentioning

confidence: 99%

Differences between perfect powers : prime power gaps

Bennett¹,

Siksek²

2021

Preprint

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“…To show that log |Λ| here is indeed small, we first require an upper bound upon the exponents α q in equation (11). From (24), we have that (35) 2…”

Section: Equation (2) With Y Even : Large Exponentsmentioning

confidence: 99%

“…By way of example, in cases (i) and (ii), equation (1), for fixed n, reduces to finitely many Thue or Thue-Mahler equations, respectively. These can be solved through arguments of Tzanakis and de Weger [34], [35], [36] (see also [15] for recent refinements).…”

Section: Introductionmentioning

confidence: 99%

Differences between perfect powers : the Lebesgue-Nagell Equation

Bennett¹,

Siksek²

2021

Preprint

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We develop a variety of new techniques to treat Diophantine equations of the shape x 2 + D = y n , based upon bounds for linear forms in p-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers x and y, and n ≥ 3, with the property that y n > x 2 and x 2 − y n has no prime divisor exceeding 11.

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“…To give the reader an idea of our running times, we now discuss some equations appearing in the literature. We solved the equation of Tzanakis-de Weger [TdW91], and we deter-mined all solutions of the equation of Agraval-Coates-Hunt-van der Poorten [ACHvdP80].…”

Section: Applicationsmentioning

confidence: 99%

“…(1.5). To compare at least some data, we solved the equations of [ACHvdP80, TdW91,TdW92] and in all cases it turned out that we found the same set of solutions.…”

Section: Comparison Of Algorithmsmentioning

confidence: 99%

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

Känel¹,

Matschke²

2016

Preprint

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In the first part we construct algorithms (over Q) which we apply to solve S-unit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for S-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures and questions, including Baker's explicit abc-conjecture and a new conjecture on S-integral points of any hyperbolic genus one curve over Q.In the second part we establish new results for certain old Diophantine problems (e.g. the difference of squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and Vinogradov-Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental cases. We also conduct some effort to work out optimized height bounds for S-unit and Mordell equations which are used in our algorithms of the first part. Our results and algorithms all ultimately rely on the method of Faltings (Arakelov, Paršin, Szpiro) combined with the Shimura-Taniyama conjecture, and they all do not use lower bounds for linear forms in (elliptic) logarithms.In the third part we solve the problem of constructing an efficient sieve for the S-integral points of bounded height on any elliptic curve E over Q with given Mordell-Weil basis of E(Q). Here we combine a geometric interpretation of the known elliptic logarithm reduction (initiated by Zagier) with several conceptually new ideas. The resulting "elliptic logarithm sieve" is crucial for some of our algorithms of the first part. Moreover, it considerably extends the class of elliptic Diophantine equations which can be solved in practice: To demonstrate this we solved many notoriously difficult equations by combining our sieve with known height bounds based on the theory of logarithmic forms.

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Solving a specific Thue-Mahler equation

Cited by 16 publications

References 8 publications

Differences between perfect powers : prime power gaps

Differences between perfect powers : prime power gaps

Differences between perfect powers : the Lebesgue-Nagell Equation

Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture

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