Sums of two $S$-units via Frey-Hellegouarch curves

Bennett, Michael A.; Billerey, Nicolas

doi:10.1090/mcom/3129

Cited by 8 publications

(8 citation statements)

References 21 publications

(77 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above conjecture has been verified in some special cases (see [1,[10][11][12][13]15], [23] and [26]). But, in general, it is far from solved problem.…”

Section: Introductionmentioning

confidence: 79%

See 1 more Smart Citation

A modular approach to the generalized Ramanujan-Nagell equation

Mutlu¹,

Mao-hua²,

Soydan³

2021

Preprint

View full text Add to dashboard Cite

Let k be a positive integer. In this paper, using the modular approach, we prove that if k ≡ 0 (mod 4), 30 < k < 724 and 2k − 1 is an odd prime power, then the equation ( * ) x 2 + (2k − 1) y = k z has only one positive integer solution (x, y, z) = (k − 1, 1, 2). The above results solve some difficult cases of Terai's conecture concerning the equation ( * ). 2020 Mathematics Subject Classification. 11D61.

show abstract

“…The above conjecture has been verified in some special cases (see [1,[10][11][12][13]15], [23] and [26]). But, in general, it is far from solved problem.…”

Section: Introductionmentioning

confidence: 79%

“…But, his proof is incorrect. Until 2017, M. A. Bennett and N. Billery [1] used the modular approach to solve the case k ∈ {12, 24}. It follows that the case 4 | k and 2k − 1 is an odd prime power is a very difficult case to Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

A modular approach to the generalized Ramanujan-Nagell equation

Mutlu¹,

Mao-hua²,

Soydan³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This class of equations contains in particular all equations considered in this paper. Here we mention that the possibility of solving certain cubic Thue-Mahler equations via modular symbols was already discussed in Bennett-Dahmen [BD13,§14], see also the recent works of Kim [Kim16] and Bennett-Billerey [BB16]. Our first type of algorithms is very fast for "small" parameters, since in this case we can use Cremona's database listing all elliptic curves over Q of conductor at most 350000 (as of August 2014).…”

Section: Algorithms Via Modular Symbolsmentioning

confidence: 99%

“…Suppose that l ∈ {2, 3}. In a recent work, Bennett-Billerey [BB16] show in particular how to practically solve the following problem (in which l = 2 is the original problem)…”

Section: Generalizedmentioning

confidence: 99%

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

Känel¹,

Matschke²

2016

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…This gives a bound on z independent of c. Recently, R. Scott and R. Styer [157] proved that if c ≡ 1 mod 2 then z < 1 2 ab. In some cases, finding a bound on z independent of c can be taken quite a bit further: when every prime dividing ab is in a given finite set of primes S, it is often possible to show that (1.1) implies z = 1 except for a finite list of specified exceptions; to do this, elementary methods sometimes suffice (e. g., S = {2, 3, 5}), but in general they do not, particularly when 2 ∈ S. M. A. Bennett and N. Billerey [3] deal with the case S = {3, 5, 7}, completely handling not only (1.1) but also the more general case in which A and B are S-units such that A + B = c z , z > 1.…”

Section: Introductionmentioning

confidence: 99%

A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

Le,

Scott,

Styer

2018

Preprint

View full text Add to dashboard Cite

show abstract

Sums of two $S$-units via Frey-Hellegouarch curves

Cited by 8 publications

References 21 publications

A modular approach to the generalized Ramanujan-Nagell equation

A modular approach to the generalized Ramanujan-Nagell equation

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

A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

Contact Info

Product

Resources

About